# Stone duality for lattice expansions

Stone duality for lattice expansions Abstract The Stone duality for bounded lattices by this author, with J.M. Dunn, is lifted in this article to a duality for lattices with operators. The dual frames of lattice expansions are two-sorted frames (X, $${\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}$$, Y), further equipped with an (n + 1)-ary relation $$\textit{R}_{\delta }$$ and a dual relation $$\textit{R}^{\partial }_{\delta }$$ for each n-ary lattice operator of some distribution type $$\delta$$. The closures of the generalized image operators generated by the relations are shown to be precisely the $$\sigma$$-extensions of the corresponding lattice operators and thus the full complex algebra of Galois-stable sets of the frame constitutes a concrete canonical extension of the lattice expansion. Thereby, the results presented in this article extend to the non-distributive case the classical Jónsson–Tarski results for Boolean algebras with operators and their extension to mere distributive lattices with operators. Consequently, the duality based approach to relational logic semantics is extended here to the case of logics dropping distribution. As an application example, we model the full BCK calculus. Both plain Kripke-type (two-sorted) frame semantics, as well as general (two-sorted) frame semantics are presented, the distinctive feature of the latter choice being that the interpretation of additional lattice operators (such as modal operators) is typically verbatim the same as in the distributive case, which is desirable in intended applications (such as temporal, or dynamic extensions of non-distributive lattice logic). 1 Introduction This article lies in the tradition of Stone dualities for lattice-based algebras [31–33] and its related set-theoretic (relational) semantics tradition for the associated logical calculi, initiated by the classical Jónsson–Tarski results on Boolean algebras with operators [23] and extended to the case of a mere distributive lattice by Urquhart [36] and others. The article builds on the theory of canonical extensions [9, 10] and on this author’s work, both past [11, 12, 19] and recent [13–17] and it constitutes a generalisation and abstraction over a recent article [20]. In [24, 25] Moshier and Jipsen studied a representation of lattices with operators, extended to a duality. Unlike [24, 25], the focus of this article is on the relational representation of normal operators on bounded lattices. The representation is extended to a full functorial duality, obtained by lifting an existing duality [19] by this author, with J.M. Dunn, for the category of bounded lattices. In [14], the same objectives were addressed, based on a duality for lattices with additional operators [12] and resulting in the development of the framework of order-dual, or Kripke–Galois semantics for non-distributive logics. We take as our starting point here the lattice representation [19], which constitutes a canonical extension of the lattice, as shown in [9], Proposition 2.6. In as far as applications in logic are concerned, our results deliver a 2-sorted version of the Kripke–Galois semantics framework of [2, 14]. The semantics of non-distributive logics has been also recently addressed by various other authors [4, 5, 7, 8, 34], invariably leading to non-standard interpretations of familiar logical operators (such as diamonds and boxes), which appears to be necessary when working with plain Kripke-type frame semantics. Our results serve as a basis both for plain Kripke-type frame semantics, as well as for a semantics based on general frames. In the latter case, the received (intended) interpretation of operators is recaptured, which is desirable in some applied contexts, such as temporal, or dynamic extensions of non-distributive lattice logic. The present article is also related to and it constitutes an advancement of Dunn’s theory of generalized Galois logics [6] and it sheds light on the semantics of substructural logics, specifically for the case where distribution of conjunction over disjunction and conversely is not assumed in the logic. For the reader’s convenience, we gather some technical preliminaries in Section 2. Specifically, Section 2.4 recalls the basic notions from the theory of canonical extensions [9, 10] and it gives a brief presentation of the lattice representation of [19]. Section 2.5 recalls the construction in a canonical lattice extension of the $$\sigma /\pi$$-extensions of lattice operators [9]. Section 2.1 presents the elementary properties of polarities (X, $${\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}$$, Y) that will be useful in the sequel, where X, Y are nonempty sets and $${\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}$$ ⊆ X × Y. Section 2.3 defines the category L$$_{\tau }$$ of normal lattice expansions of some similarity type $$\tau$$. Section 3 defines the category of $$\tau$$-frames, in steps. First, in Section 3.1 we consider base frames (polarities) with relations, we define the generated generalized image operators and we prove that they restrict to operators on the families of closed and of clopen sets of their respective domains and that their dual operators can be obtained from the duals of the frame relations, which we define in a canonical way. Section 2.2 reviews the definition of lattice frames and their morphisms, from the lattice representation of [19]. It is in Section 3.2 that objects of the category of $$\tau$$-frames are axiomatized and their morphisms are defined. Finally, Section 4 is devoted to the proof that the duality of the categories of bounded lattices and ⊥-frames proven in [19] lifts to a duality of the categories of normal lattice expansions of similarity type $$\tau$$ with the corresponding categories of $$\tau$$-frames. Much of Section 4 is devoted to the construction of the dual frame of a lattice expansion $${\mathcal A}_{\tau }=({\mathcal L},(\,f_{\delta })_{\delta \in \tau _{1}},(h_{\delta })_{\delta \in \tau _{\partial }})$$. To define the canonical relations on the frame we make use of the point operators we introduced in [12]. The key point in the argument is the proof that the restriction of a generated set-operator on the closed sets of its domain is the $$\sigma$$-extension of its restriction to clopen sets. Full functorial duality is proven in Theorem 4.5. Section 5 presents an application of the developed framework, modeling full BCK. For the reader’s benefit, we present both a plain Kripke-type semantics for BCK, as well as a semantics based on general frames, both using the representation results of this article. 2 Technical preliminaries 2.1 Polarities A polarity (base frame) is a triple (X, $${\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}$$, Y) where X, Y are nonempty sets (of worlds and co-worlds) and $${\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}$$ ⊆ X × Y is a binary relation, to be called the Galois relation of the frame, generating a Galois connection defined [3] on U ⊆ X and V ⊆ Y by   \begin{align*} \phi(U)=U^{{\mathop{=}^{\kern-5pt\shortmid}}} =&\{y\in Y\;|\;\forall u\in U\; u\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; y\}=\{y\in Y\;|\; U\ {\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\ y\}\\ \psi(V)={}^{{\mathop{=}^{\kern-5pt\shortmid}}} V =&\{x\in X\;|\; \forall v\in V\; x\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; v\}=\{x\in X\;|\; x\ {\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\ V\}. \end{align*}A subset A ⊆ X is Galois-stable if $$A=\psi \phi (A)$$ and we let $${\mathcal G}_{\psi }(X)$$ be the complete lattice of Galois-stable subsets of X. Similarly for $${\mathcal G}_{\phi }(Y)$$ and the complete lattice of co-stable subsets of Y, $$B=\phi \psi (B)$$. We also let $$\varnothing _{\psi },\varnothing _{\phi }$$ be the least elements of $${\mathcal G}_{\psi }(X)$$ and $${\mathcal G}_{\phi }(Y)$$, respectively, i.e. the intersections of all their members, and we note that they need not be empty. The relations x ≤ z iff $$\{x\}^{{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}} }\subseteq \{z\}^{{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}} }$$ on X and y ≤ v iff $${ }^{{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}} }\{y\}\subseteq{ }^{{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}} }\{z\}$$ on Y are preorders on X and Y, respectively. We make the further assumptions that (F1) X, Y are both partial orders under their respective ≤-relation (F2) $${\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}$$ is increasing in each argument place, i.e. x$${\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}$$y, x ≤ z, y ≤ v imply z$${\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}$$v. Note that the partial-ordering assumption implies that base frames are separated (S-frames), in the sense of [8]. We use $$\Gamma$$ for the upper closure operator, $$\Gamma U=U^{\uparrow }$$. In particular, for x ∈ X (resp. y ∈ Y) we write $$\Gamma x$$ for the principal upper set over x, as shorthand for the more accurate $$\Gamma (\{x\})$$: $$\Gamma x=\{z\in X\;|\; x\leq z\}$$. Similarly for $$\Gamma y$$, with y ∈ Y. Lemma 2.1 Let (X, $${\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}$$, Y) be a base frame. Then the following hold, for any x ∈ X, y ∈ Y, (1) $$(\Gamma x){ }^{{\mathop{=}^{\kern-5pt\shortmid}}}=\{x\}{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}$$ and $${ }^{{\mathop{=}^{\kern-5pt\shortmid}}}(\Gamma y)={ }^{{\mathop{=}^{\kern-5pt\shortmid}}}\{y\}$$ (2) $$\Gamma x\in \mathcal{G}_{\psi }(X)$$ and $$\Gamma y\in \mathcal{G}_{\phi }(Y)$$. Hence also $${ }^{{\mathop{=}^{\kern-5pt\shortmid}}}\{y\}\in \mathcal{G}_{\psi }(X)$$ and $$\{x\}{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}\in \mathcal{G}_{\phi }(Y)$$ (3) Stable and co-stable sets are increasing, i.e. u ∈ A implies $$\Gamma u\subseteq A$$ (4) For any $$A\in \mathcal{G}_{\psi }(X)$$, $$A=\bigvee _{x\in A}\Gamma x=\bigcap _{A {\mathop{=}^{\kern-5pt\raise-4pt\shortmid}} y}({ }^{{\mathop{=}^{\kern-5pt\shortmid}}}\{y\})$$ and similarly for $$B\in{\mathcal G}_{\phi }(Y)$$. Proof. For 1), left-to-right is immediate and the other direction uses increasingness of $${\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}$$. For 2), use 1) and the definition of the partial order. For 3) we do only the case for stable sets, since the argument for co-stable sets is completely similar. We have $$x\in A\Longrightarrow \{x\}\subseteq A\Longrightarrow A{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}\subseteq \{x\}{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}=(\Gamma x){ }^{{\mathop{=}^{\kern-5pt\shortmid}}}$$ , using claim 1). Given that $$\Gamma x$$ is stable, by claim 2), we obtain $$\Gamma x={ }^{{\mathop{=}^{\kern-5pt\shortmid}}}((\Gamma x){ }^{{\mathop{=}^{\kern-5pt\shortmid}}})\subseteq{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}(A{ }^{{\mathop{=}^{\kern-5pt\shortmid}}})=A$$. 4) now follows, since clearly $$A\subseteq \bigvee _{x\in A}\Gamma x$$ and, conversely, $$x\in A\Longrightarrow \Gamma x\subseteq A$$, using claim 3), hence $$\bigvee _{x\in A}\Gamma x\subseteq A$$, as well. Similarly for $$B\in{\mathcal G}_{\phi }(Y), B=\bigvee _{y\in B}\Gamma y$$. Then $$A{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}=\bigvee _{y\in A{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}}\Gamma y$$, hence applying $$\psi$$ we obtain $$A=\bigcap _{A\subseteq{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}\{y\}}({}^{{\mathop{=}^{\kern-5pt\shortmid}}}\{y\})=\bigcap _{A\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; y}({ }^{{\mathop{=}^{\kern-5pt\shortmid}}}\{y\})$$. Let $${\mathcal G}_{\kappa }(X)=\{\Gamma x\;|\; x\in X\}$$ and $${\mathcal G}_{o}(X)=\{{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}\{y\}\;|\; y\in Y\}$$ and similarly for $${\mathcal G}_{\kappa }(Y),{\mathcal G}_{o}(Y)$$. It has been shown above that Corollary 2.2 $${\mathcal G}_{\kappa }(X)$$ is join-dense in $${\mathcal G}_{\psi }(X)$$, while $${\mathcal G}_{o}(X)$$ is meet-dense in $${\mathcal G}_{\psi }(X)$$ and similarly for $${\mathcal G}_{\phi }(Y)$$. Definition 2.3 (Closed and open elements) The closed (filter) elements of $${\mathcal G}_{\psi }(X)$$ are the join-generators $$\Gamma x$$, with x ∈ X, and its open (ideal) elements are the meet-generators $${ }^{{\mathop{=}^{\kern-5pt\shortmid}}}\{y\}$$, with y ∈ Y. Similarly for $${\mathcal G}_{\phi }(Y)$$. A stable set A is a clopen element of $${\mathcal G}_{\psi }(X)$$ if it is both closed and open, i.e. $$\Gamma x=A={ }^{{\mathop{=}^{\kern-5pt\shortmid}}}\{y\}$$, for some x ∈ X, y ∈ Y, both necessarily unique. Similarly for clopen elements $$\{x\}{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}=B=\Gamma y$$ of $${\mathcal G}_{\phi }(Y)$$. We let $${\mathcal G}_{\kappa o}$$ designate clopen elements. By a slight abuse of terminology, a point x ∈ X will be called clopen when $$\Gamma x$$ is clopen and similarly for y ∈ Y. Note that, since reversing the order switches meets and joins, the closed elements of $${\mathcal G}_{\psi }(X)^{\partial }$$ are the ones that are open in $${\mathcal G}_{\psi }(X)$$, i.e. the elements $${ }^{{\mathop{=}^{\kern-5pt\shortmid}}}\{y\}$$, while its open elements are, dually, the closed elements of $${\mathcal G}_{\psi }(X)$$. Remark 2.4 (Closed elements of products) Up to isomorphism, the closed elements of a product are tuples of closed elements of the factors of the product and similarly for open elements. Since we shall often have use of closed and open elements of products, it is useful to point out that in e.g. $${\mathcal G}_{\psi }(X)\times{\mathcal G}_{\psi }(X)^{\partial }$$ a closed element is a pair $$(K,K^{\prime })$$ where K is closed in $${\mathcal G}_{\psi }(X)$$ and $$K^{\prime }$$ is closed in $${\mathcal G}_{\psi }(X)^{\partial }$$, hence open in $${\mathcal G}_{\psi }(X)$$. It is therefore a pair of the form $$(\Gamma x,{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}\{y\})$$. More generally, if $$\delta =(i_{1},\ldots ,i_{n};i_{n+1})$$ is a distribution type of a normal lattice operator f, we will be constructing maps $$F:{\mathcal G}_{\psi }(X)^{i_{1}}\times \cdots \times{\mathcal G}_{\psi }(X)^{i_{n}}\longrightarrow{\mathcal G}_{\psi }(X)^{i_{n+1}}$$, defined on closed elements of the product, i.e. tuples of the form $$(\ldots ,\underbrace{\Gamma u_{j}}_{i_{j}=1},\ldots ,\underbrace{{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}\{u_{r}\}}_{i_{r}=\partial },\ldots )$$, where for $$i_{j}=1,\; u_{j}\in X$$, while for $$i_{r}=\partial ,\;u_{r}\in Y$$. Dually, an open element of the product is a tuple of the form $$(\ldots ,\underbrace{{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}\{ u_{j}\}}_{i_{j}=1},\ldots ,\underbrace{\Gamma u_{r}}_{i_{r}=\partial },\ldots )$$, where for $$i_{j}=1,\; u_{j}\in Y$$, while for $$i_{r}=\partial ,\;u_{r}\in X$$. The Galois maps $$\psi ={ }^{{\mathop{=}^{\kern-5pt\shortmid}}}(\;),\phi =(\;){ }^{{\mathop{=}^{\kern-5pt\shortmid}}}$$ provide a dual equivalence $${\mathcal G}_{\psi }(X)^{\partial }\backsimeq{\mathcal G}_{\phi }(Y)$$. It is also useful to work with two-sorted products whose factors are either $${\mathcal G}_{\psi }(X)$$ or $${\mathcal G}_{\phi }(Y)$$. If $$\delta =(i_{1},\ldots ,i_{n};i_{n+1})$$, we shall also be considering products $${\mathcal G}(Z^{i_{1}})\times \cdots \times{\mathcal G}(Z^{i_{n}})\longrightarrow{\mathcal G}(Z^{i_{n+1}})$$, where $${\mathcal G}(Z^{i_{j}})$$ is $${\mathcal G}_{\psi }(X)$$ if $$i_{j}=1$$ and it is $${\mathcal G}_{\phi }(Y)$$ if $$i_{j}=\partial$$. Closed elements of produts are then tuples $$(\Gamma u_{1},\ldots ,\Gamma u_{n})$$, where $$\Gamma u_{j}\in{\mathcal G}_{\psi }(X)$$ (for some $$u_{j}\in X$$) if $$i_{j}=1$$ and $$\Gamma u_{r}\in{\mathcal G}_{\phi }(Y)$$ (for some $$u_{r}\in Y$$) if $$i_{r}=\partial$$. 2.2 Lattice frames Lattice frames (X, $${\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}$$, Y) have been defined in [19], involving topologies on each of X, Y. We will therefore avoid re-inventing the wheel and we shall base our definition on pre-existing work. For a lattice frame we require in addition to axioms (F1,F2) for base frames that (F3) Clopen sets are closed under finite intersections and closed sets are closed under arbitrary intersections, for each of $${\mathcal G}_{\psi }(X),{\mathcal G}_{\phi }(Y)$$. (F4) If A is an index set for clopen elements $$\Gamma x_{a}={ }^{{\mathop{=}^{\kern-5pt\shortmid}}}\{y_{a}\}$$, the set $$\{\Gamma x_{a}\;|\; a\in A\}\cup \{-\Gamma x_{a}\;|\; a\in A\}$$ is a subbasis for a compact, totally separated topology on X. Similarly for Y. In other words, X, Y are both Stone spaces. (F5) The family of closed sets is the meet-closure of the family of clopen sets, for each of $${\mathcal G}_{\psi }(X),{\mathcal G}_{\phi }(Y)$$. Axioms (F3–F5) are equivalent to the conditions defining FSpaces in [19]. By choice of the subbasis, clopens in the algebraic sense are precisely the clopen sets in the topological sense. The axioms clearly imply that clopens form a lattice, with joins defined by $$\Gamma x_{a}\vee \Gamma x_{b}=\psi (\phi \Gamma x_{a}\cap \phi \Gamma x_{b})=\psi (\{x_{a}\}{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}\cap \{x_{b}\}{ }^{{\mathop{=}^{\kern-5pt\shortmid}}})=\psi (\Gamma y_{a}\cap \Gamma y_{b})$$. They also imply that $${\mathcal G}_{\kappa }(X)$$ is a complete meet-subsemillatice of $${\mathcal G}_{\psi }(X)$$, hence a complete lattice (but arbitrary joins do not necessarily coincide in $${\mathcal G}_{\kappa }(X),{\mathcal G}_{\psi }(X)$$). This induces a complete lattice structure on X (and similarly for Y). Furthermore, every closed element $$\Gamma x$$ is obtained as the meet of the clopens that cover it, i.e. $$\Gamma x=\bigwedge \{\Gamma x_{a}\;|\;\Gamma x\subseteq \Gamma x_{a}\}$$ and similarly for $${\mathcal G}_{\kappa }(Y)$$. Consequently, clopens are meet-dense in the lattice of closed elements. Furthermore, meet-density of clopens in $${\mathcal G}_{\phi }(Y)$$ evidently implies that clopens are also join-dense in $${\mathcal G}_{\kappa }(X)$$ (and similarly for $${\mathcal G}_{\kappa }(Y)$$ and $${\mathcal G}_{\phi }(Y)$$). Lemma 2.5 (1) Each of X, Y is an FSpace, in the sense of [19]. (2) The frame (X, $${\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}$$, Y) is a ⊥-frame, in the sense of [19]. (3) $$(\imath ,{\mathcal G}_{\psi }(X))$$ is a canonical extension of the lattice $${\mathcal G}_{\kappa o}(X)$$ of clopen elements of $${\mathcal G}_{\psi }(X)$$, where ı is the inclusion of clopens into $${\mathcal G}_{\psi }(X)$$, and $$(\jmath ,{\mathcal G}_{\phi }(Y))$$ is a dual canonical extension, where $$\jmath (\Gamma x_{a})=\{x_{a}\}{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}=\Gamma y_{a}$$ is the inclusion map of clopens into $${\mathcal G}_{\phi }(Y)$$. Proof. The proof for 1) and 2) is immediate, by consulting the definitions in [19]. For 3), combine with the results of [9], where it is shown that the representation of [19] delivers a canonical extension. Definition 2.6 (Frame morphisms, [19])A morphism$$(\,f,h):(X_{1},{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}} _{1},Y_{1})\longrightarrow (X_{2},{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}} _{2},Y_{2})$$of lattice frames is a pair of functions $$f:X_{1}\longrightarrow X_{2},\;h:Y_{1}\longrightarrow Y_{2}$$, such that f, h are continuous functions that preserve greatest lower bounds in $$X_{1},Y_{1}$$, respectively, and whose inverse image takes clopens to clopens (called FSpace-morphisms in [19]). Setting $$\,f^{\ast }=f^{-1}, h^{\ast }=h^{-1}$$, both squares in the diagram below commute where $$X_{i}^{\ast },Y_{i}^{\ast }$$ stand for the respective collections of clopen sets. 2.3 Normal lattice expansions This section introduces categories of algebraic structures of interest in the present article, i.e. expansions of bounded lattices by normal operators, typically arising as the Lindenbaum–Tarski algebras of logical calculi. By a distribution type we mean an element $$\delta$$ of the set $$\{1,\partial \}^{n+1}$$, for some n ≥ 0, typically to be written as $$\delta =(i_{1},\ldots ,i_{n};i_{n+1})$$ and where $$\delta _{(n+1)}=i_{n+1}\in \{1,\partial \}$$ will be referred to as the output type of $$\delta$$. A similarity type$$\tau$$ is then defined as a finite sequence of distribution types, $$\tau =\langle \delta _{1},\ldots ,\delta _{k}\rangle$$. Definition 2.7 (Normal operators) Following [23], an n-ary monotone operator $$f:{\mathcal L}^{n}\longrightarrow{\mathcal L}$$ will be called additive if it distributes over joins of $$\mathcal L$$ in each argument place. More generally, if $${\mathcal L}_{1},\ldots ,{\mathcal L}_{n},{\mathcal L}$$ are bounded lattices, then a monotone function $$f:{\mathcal L}_{1}\times \cdots \times{\mathcal L}_{n}\longrightarrow{\mathcal L}$$ is additive, if for each i, f distributes over binary joins of $${\mathcal L}_{i}$$, i.e. $$f(a_{1},\ldots ,a_{i-1},b\vee d,a_{i+1},\ldots ,a_{n})=f(a_{1},\ldots ,a_{i-1},b,a_{i+1},\ldots ,a_{n})\vee f(a_{1},\ldots ,a_{i-1},d,a_{i+1},\ldots ,a_{n})$$. As a matter of notation, we write $$\mathcal L$$ for $${\mathcal L}^{1}$$ and $${\mathcal L}^{\partial }$$ for its opposite lattice (where order is reversed, usually designated as $${\mathcal L}^{op}$$). An n-ary operator f on a lattice $$\mathcal L$$ is normal [12] if it is an additive function $$f:{\mathcal L}^{i_{1}}\times \cdots \times{\mathcal L}^{i_{n}}\longrightarrow{\mathcal L}^{i_{n+1}}$$, where each $$i_{j}$$, for $$j=1,\ldots ,n,n+1$$, is in the set {1, ∂}, i.e. $${\mathcal L}^{i_{j}}$$ is either $$\mathcal L$$, or $${\mathcal L}^{\partial }$$. For a normal operator f on $$\mathcal L$$, its distribution type is the $$(n+1)$$-tuple $$\delta (f)=(i_{1},\ldots ,i_{n};i_{n+1})$$. We call f completely normal if it (co)distributes over arbitrary joins, or meets, at each argument place. Definition 2.8 A lattice expansion is a structure $${\mathcal L}=(L,\wedge ,\vee ,0,1,(\,f_{i})_{i\in k})$$ where k > 0 is a natural number and for each i ∈ k, $$\;f_{i}$$ is a normal operator on $$\mathcal L$$ of some specified arity $$\alpha (\,f_{i})\in \mathbb{N}^{+}$$ and distribution type $$\delta (i)$$. The similarity type of $$\mathcal L$$ is the k-tuple $$\tau ({\mathcal L})=\langle \delta (0),\ldots ,\delta (k-1)\rangle$$. Where $$\tau$$ is a similarity type, $$\mathbb{L}_{\tau }$$ is the variety of lattice expansions of similarity type $$\tau$$. L$$_{\tau }$$ designates the category of lattice expansions of similarity type $$\tau$$. Morphisms of lattice expansions are the bound preserving homomorphisms in the usual algebraic sense. Example 2.9 A bounded lattice with a box and a diamond operator $${\mathcal L}=(L,\leq ,\wedge ,\vee ,0,1,\Box ,\Diamond )$$ is an object of L$$_{\tau }$$, where $$\tau$$ is the similarity type $$\tau =\langle (1;1),(\partial ,\partial )\rangle$$ where $$\delta (\Diamond )=(1;1)$$, i.e. $$\Diamond :{\mathcal L}\longrightarrow{\mathcal L}$$ distributes over joins of $$\mathcal L$$, while $$\delta (\Box )=(\partial ;\partial )$$, i.e. $$\Box :{\mathcal L}^{\partial }\longrightarrow{\mathcal L}^{\partial }$$ distributes over ‘joins’ of $${\mathcal L}^{\partial }$$ (i.e. meets of $$\mathcal L$$), delivering ‘joins’ of $${\mathcal L}^{\partial }$$ (i.e. meets of $$\mathcal L$$). An implicative lattice is an object of L$$_{\tau ^{\prime }}$$, where $$\tau ^{\prime }=\langle (1,\partial ;\partial )\rangle$$ and where $$(1,\partial ;\partial )=\delta (\rightarrow )$$ is the distribution type of the implication operator, regarded as a map $$\rightarrow \;:{\mathcal L}\times{\mathcal L}^{\partial }\longrightarrow{\mathcal L}^{\partial }$$ distributing over ‘joins’ in each argument place, i.e. co-distributing over joins in the first place, turning them to meets, and distributing over meets (joins of $${\mathcal L}^{\partial }$$) in the second place, delivering ‘joins’ of $${\mathcal L}^{\partial }$$, i.e. meets of $$\mathcal L$$. An FL-algebra (Full Lambek algebra [27]) is an object of L$$_{\tau ^{^{\prime \prime }}}$$, where $$\tau ^{^{\prime \prime }}=\langle (1,1;1),(1,\partial ;\partial ),(\partial ,1;\partial )\rangle$$, i.e. an algebra $${\mathcal L}=(L,\leq ,\wedge ,\vee ,0,1,\leftarrow ,\circ ,\rightarrow )$$, with $$\delta (\leftarrow )=(\partial ,1;\partial ), \delta (\circ )=(1,1;1)$$ and $$\delta (\rightarrow )=(1,\partial ;\partial )$$. Definition 2.10 A canonical extension of a lattice expansion$$(L,\wedge ,\vee ,0,1,(\,f_{i})_{i\,\in\, k})$$ is a canonical lattice extension $$(\alpha ,C)$$ [9] (see Section 2.4 for a brief review) for the underlying bounded lattice together with an n-ary operator $$F_{i}$$, corresponding to the lattice operator $$f_{i}$$ such that in each argument place if $$f_{i}$$ (co)distributes over finite joins (or meets), then $$F_{i}$$ (co)distributes over arbitrary joins (resp. meets). It is shown in [9] that canonical extensions of normal lattice expansions exist, by constructing extensions $$f_{\sigma }=f_{\pi }$$ (identity follows from the normality assumption for f), as detailed in Section 2.5. 2.4 Canonical extensions For the reader’s convenience, we briefly review in this and the next section the basics on canonical extensions of bounded lattices and $$\sigma /\pi$$-extensions of maps. In [9] a notion of canonical extension of bounded lattices was introduced, generalising the corresponding notion for distributive lattices and Boolean algebras [10] and which characterizes the dual objects of lattices in purely lattice-theoretic terms, without resorting to topological properties. A canonical extension of a bounded lattice $$\mathcal L$$ is defined in [9] as a pair $$(\alpha ,C)$$, where C is a complete lattice and $$\alpha :{\mathcal L}\hookrightarrow C$$ is a lattice embedding and where (density) $$\alpha [{\mathcal L}]$$ is dense in C, where the latter means that every element of C can be expressed both as a meet of joins and as a join of meets of elements in $$\alpha [{\mathcal L}]$$ (compactness) for any set A of closed elements and any set B of open elements of C, $$\bigwedge A\leq \bigvee B$$ iff there exist finite subcollections $$A^{\prime }\subseteq A, B^{\prime }\subseteq B$$ such that $$\bigwedge A^{\prime }\leq \bigvee B^{\prime }$$ where the closed elements of C are defined in [9] as the elements in the meet-closure of the representation map $$\alpha$$ and the open elements of C are defined dually as the join-closure of the image of $$\alpha$$. In other words, if $$\alpha$$ is the representation map and $$M\subseteq{\mathcal L}$$, then $$\bigwedge \{\alpha (a)\;|\; a\in M\}$$ defines a closed element, while $$\bigvee \{\alpha (a)\;|\; a\in M\}$$ defines, dually, an open element. In [9], Proposition 2.6, existence of canonical extensions for bounded lattices is proven by showing that the completion of a bounded lattice $$\mathcal L$$ obtained in the lattice representation theorem of [18, 19] by this author and J.M. Dunn is a canonical extension of $$\mathcal L$$. Furthermore, canonical extensions are proven to be unique, up to isomorphism ([9], Proposition 2.7). Urquhart’s [35] and, subsequently, Hartung’s [22] lattice representations (both predating [18, 19]) also constitute canonical extensions of the represented lattice (see [9], though no proof is presented), as do the representations due to Ploščica [30] and to Allwein and Hartonas [1]. Using Urquhart’s representation, in particular, it is easily proven (see [35]) that the representation reduces to the Priestley representation [31] when the represented lattice is distributive. Our approach in the present article is based on the representation [18, 19] by this author, with J.M. Dunn. In [18] representation and duality theorems for partial orders, semilattices and Galois connections were proven and the lattice representation and duality of [19] is obtained by representing the trivial duality $${\mathcal L}\backsimeq ({\mathcal L}^{\partial })^{\partial }$$, via the identity map. If $${\mathcal L}$$ is a bounded lattice then $${\mathcal L}_{+}=\mathfrak{F}=(X,{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}} , Y)$$, following [19], is the canonical dual frame of the lattice $${\mathcal L}$$, where $$X=\textrm{Filt}({\mathcal L})$$ is the set of lattice filters, $$Y=\textrm{Idl}({\mathcal L})$$ is the set of lattice ideals and where $${\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}$$ ⊆ X × Y represents the trivial (identity) duality $${\mathcal L}\backsimeq ({\mathcal L}^{\partial })^{\partial }$$, explicitly defined by x$${\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}$$y iff ∃a ∈ x i(a) ∈ y iff $$x\cap y\neq \emptyset$$. The relation $${\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}$$, which we refer to as the Galois relation of the frame, generates a Galois connection [3]. It is proven in [9] that $${\mathcal L}^{\sigma }=\{A\subseteq X\;|\; A=\psi \phi (A)\}={\mathcal G}_{\psi }(X)$$ is a canonical extension of $$\mathcal L$$ (dually isomorphic to $$({\mathcal L}^{\sigma })^{\partial }=\{B\subseteq Y\;|\; B=\phi \psi (B)\}={\mathcal G}_{\phi }(Y)$$). The canonical (co)representation maps are given by $$\alpha _{X}(a)=\alpha (a)=\{x\in X\;|\; a\in x\}$$ and $$\alpha _{Y}(a)=\beta (a)=\{y\in Y\;|\; a\in y\}$$. 2.5 $$\sigma ,\pi$$-Extensions of lattice maps If $$(\alpha ,C)$$ is a canonical extension of a bounded lattice $$\mathcal L$$, and K, O are its sets of closed and open elements, the $$\sigma$$ and $$\pi$$-extensions $$f_{\sigma },f_{\pi }:{\mathcal L}_{\sigma }\longrightarrow{\mathcal L}_{\sigma }$$ (where, following the notation of [9], $${\mathcal L}_{\sigma }={\mathcal G}_{\psi }(X)$$ designates the canonical extension of $$\mathcal L$$) of a unary monotone map $$f:{\mathcal L}\longrightarrow{\mathcal L}$$ are defined in [9], taking also into consideration Lemma 4.3 of [9], by setting, for k ∈ K, o ∈ O and u ∈ C  \begin{align} f_{\sigma}(k)=\bigwedge\{\,f(a)\;|\; k\leq a\in L\} \qquad f_{\sigma}(u)=\bigvee\{\,f_{\sigma}(k)\;|\;{\tt K}\ni k\leq u\} \end{align} (1)  \begin{align} f_{\pi}(o)=\bigvee \{\,f(a)\;|\; L\ni a\leq o\} \qquad f_{\pi}(u)=\bigwedge\{\,f_{\pi}(o)\;|\; u\leq o\in{\tt O}\} \end{align} (2)where in these definitions $$\mathcal L$$ is identified with its isomorphic image in C and $$a\in{\mathcal L}$$ is then identified with its representation image. Remark 2.11 In [9], the authors focus on the canonical extension $${\mathcal L}^{\sigma }={\mathcal G}_{\psi }(X)$$ of $$\mathcal L$$ and the use of its dual $${\mathcal G}_{\phi }(Y)\backsimeq ({\mathcal L}^{\sigma })^{\partial }$$ is secondary, since $$({\mathcal L}^{\sigma })^{\partial }\backsimeq ({\mathcal L}^{\partial })^{\sigma }$$ and therefore $${\mathcal G}_{\phi }(Y)$$ is just the canonical extension of $${\mathcal L}^{\partial }$$. Hence they have no use of the dual $$\sigma$$-extensions of maps and they prefer to work with their images in $${\mathcal L}^{\sigma }$$, the $$\pi$$-extensions, obtained by conjugating with the dual equivalence $$\phi \!:{\mathcal G}_{\psi }(X)\!\longleftrightarrow\! {\mathcal G}_{\phi }(Y)^{\partial}\!:\psi$$. It is advantageous for our purposes to work simultaneously with both $${\mathcal G}_{\phi }(Y)$$ and $${\mathcal G}_{\psi }(X)$$ and we will systematically do so. For antitone maps, since the filters of $${\mathcal L}^{\partial }$$ are the ideals of $${\mathcal L}$$, i.e. $$\textrm{Filt}({\mathcal L}^{\partial })=\textrm{Idl}({\mathcal L})$$, and conversely $$\textrm{Idl}({\mathcal L}^{\partial })=\textrm{Filt}({\mathcal L})$$, the canonical frame for $${\mathcal L}^{\partial }$$, after [19], is the frame $$(X^{\prime },{{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}}^{\prime },Y^{\prime })=(Y,{{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}}^{-1},X)$$, where X is the set of filters of $${\mathcal L}$$ (hence the ideals of $${\mathcal L}^{\partial }$$), Y is its set of ideals (the filters of $${\mathcal L}^{\partial }$$) and where $${{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}}^{-1}\;\subseteq Y\times X$$, $$y\;{{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}}^{-1} x$$ iff x$${\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}$$y iff $$x\cap y\neq \emptyset$$. Let $$\phi ^{\prime },\psi ^{\prime }$$ be the generated Galois connection , where for V ⊆ Y, U ⊆ X we have $$\phi ^{\prime }(V)=\{x\in X\;|\; V{{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}}^{-1}x\}$$ = $$\{x\in X\;|\; x{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}} V\}=\psi (V)={ }^{{\mathop{=}^{\kern-5pt\shortmid}}} V$$ and $$\psi ^{\prime }(U)=\{y\in Y\;|\; y\;{{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}}^{-1}U\}=\{y\in Y\;|\; U\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}} \;y\}=\phi (U)= U^{{\mathop{=}^{\kern-5pt\shortmid}}}$$. Therefore, $${\mathcal G}_{\psi ^{\prime }}(X^{\prime })={\mathcal G}_{\phi }(Y)\subseteq (Y)$$ and $${\mathcal G}_{\phi ^{\prime }}(Y^{\prime })={\mathcal G}_{\psi }(X)\subseteq (X)$$. Since $${\mathcal G}_{\psi }(X)\backsimeq{\mathcal G}_{\phi }(Y)^{\partial }$$, we have $${\mathcal G}_{\psi ^{\prime }}(X^{\prime })\backsimeq{\mathcal G}_{\psi }(X)^{\partial }$$. In other words, $$(L^{\partial })_{\sigma }\backsimeq (L_{\sigma })^{\partial }$$. For n-ary maps and product lattices a similar analysis shows that $$({\mathcal L}\times{\mathcal M})_{\sigma }\backsimeq{\mathcal L}_{\sigma }\times{\mathcal M}_{\sigma }$$. Literally, the $$\sigma$$-extension of n-ary maps with mixed monotonicity properties defined in [9] is a map $$f^{\sigma }:{\mathcal G}(Z^{i_{1}})\times \cdots \times{\mathcal G}(Z^{i_{n}})\longrightarrow{\mathcal G}_{\psi }(X)$$, where $${\mathcal G}(Z^{i_{j}})=\left\{\begin{smallmatrix}{\mathcal G}_{\psi }(X) &\ \textrm{when}\ i_{j}=1\\ {\mathcal G}_{\phi }(Y) &\ \textrm{when}\ i_{j}=\partial \end{smallmatrix}\right.$$, which is defined on closed elements by $$f^{\sigma }(\Gamma u_{1},\ldots ,\Gamma u_{n})=\bigwedge \{\Gamma x_{fa_{1}\cdots a_{n}}\;|\;\bigwedge _{j}(a_{j}\in u_{j})\}$$, since $$\alpha (\,fa_{1}\cdots a_{n})=\Gamma x_{fa_{1}\cdots a_{n}}$$. Composing with the dual equivalence $$(\psi ,\phi )$$ at the appropriate argument places we get the equivalent definition of the map $$f^{\sigma }:{\mathcal G}_{\psi }(X)^{i_{1}}\times \cdots \times{\mathcal G}_{\psi }(X)^{i_{n}}\longrightarrow{\mathcal G}_{\psi }(X)$$, where $${\mathcal G}_{\psi }(X)^{i_{j}}=\left\{\begin{smallmatrix}{\mathcal G}_{\psi }(X) & \textrm{when}\ i_{j}=1\\{\mathcal G}_{\psi }(X)^{\partial } & \textrm{when}\ i_{j}=\partial \end{smallmatrix}\right.$$, defined by $$f^{\sigma }(\ldots ,\underbrace{\Gamma u_{j}}_{i_{j}=1},\ldots ,\underbrace{{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}\{u_{r}\}}_{i_{r}=\partial },\ldots )=\bigwedge \{\Gamma x_{fa_{1}\cdots a_{n}}\;|\;\bigwedge _{j}(a_{j}\in u_{j})\}=\Gamma (\bigvee \{x_{fa_{1}\cdots a_{n}}\;|\;\bigwedge _{j}(a_{j}\in u_{j})\})$$. 3 Categories of $$\tau$$-frames 3.1 Relations and operators on base frames Let $$\tau$$ be a similarity type and assume $$\tau _{1},\tau _{\partial }$$ consist of the distribution types in $$\tau$$ of output type 1 and ∂, respectively. We consider frames $$\mathfrak{F}=(X,(R_{\delta })_{\delta \in \tau _{1}},{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}} ,Y,(S^{\partial }_{\delta })_{\delta \in \tau _{\partial }})$$ where (X, $${\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}$$, Y) is a base frame in the sense of Section 2.1 and $$R_{\delta }, S^{\partial }_{\delta }$$ are relations satisfying the following conditions: (R1) If $$\delta =(i_{1},\ldots ,i_{n};i_{n+1})$$, then   \begin{align} \begin{array}{cl} (\textrm{Case}\ \delta\in\tau_{1}) & R_{\delta} \subseteq\; X\times (X^{i_{1}}\times\cdots\times X^{i_{n}})\\ (\textrm{Case}\ \delta\in\tau_{\partial}) & S^{\partial}_{\delta} \subseteq \;Y\times (X^{i_{1}}\times\cdots\times X^{i_{n}}) \end{array} \ \textrm{where}\ X^{i_{j}}\;=\;\left\{ \begin{array}{@{}cl@{}} X & \textrm{if}\ i_{j}=1\\ Y & \textrm{if}\ i_{j}=\partial. \end{array} \right. \end{align} (3) (R2) $$R_{\delta }, S^{\partial }_{\delta }$$ are increasing in the first argument place and decreasing in every other argument place. (R3) For $$\delta \in \tau _{1}$$ and any $$(u_{1},\ldots ,u_{n})\in (X^{i_{1}}\times \cdots \times X^{i_{n}})$$, the set $$R_{\delta } u_{1}\cdots u_{n}$$ is a closed element of $${\mathcal G}_{\psi }(X)$$ (in other words, there exists a point z ∈ X such that for any x ∈ X, $$xR_{\delta } u_{1}\cdots u_{n}$$ iff z ≤ x) and if all $$u_{i}$$ are clopen, then so is $$R_{\delta } u_{1}\cdots u_{n}$$. (R4) For $$\delta \in \tau _{\partial }$$ and any $$(u_{1},\ldots ,u_{n})\in (X^{i_{1}}\times \cdots \times X^{i_{n}})$$, the set $$S^{\partial }_{\delta } u_{1}\cdots u_{n}$$ is a closed element of $${\mathcal G}_{\phi }(Y)$$ (i.e. there is a point v ∈ Y such that for all y ∈ Y, $$yS^{\partial }_{\delta } u_{1}\cdots u_{n}$$ iff v ≤ y) and if all $$u_{i}$$ are clopen, then so is $$S^{\partial }_{\delta } u_{1}\cdots u_{n}$$. Definition 3.1 (Generalized image operators) The relations generate operators, as defined below   \begin{align} &(\textrm{Case}\ \delta\in\tau_{1})\quad\widehat{\bigcirc}{\kern-6.3pt\mid}\ \,_{\delta}(U_{1},\ldots,U_{n})\nonumber\\ &\quad=\{x\in X\;|\; \exists u_{1}\cdots u_{n}\;(xR_{\delta} u_{1}\cdots u_{n}\;\wedge\;\bigwedge_{j}^{i_{j}=1}(u_{j}\in U_{j})\;\wedge\;\bigwedge_{r}^{i_{r}=\partial}(U_{r}\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; u_{r}))\} \end{align} (4)  \begin{align} &(\textrm{Case}\ \delta\in\tau_{\partial})\quad\widehat{\ominus}^{\partial}_{\delta}(V_{1},\ldots,V_{n}) \nonumber\\ &\quad= \{\,y\in Y\;|\; \exists v_{1}\cdots v_{n}\;(yS^{\partial}_{\delta} v_{1}\cdots v_{n}\;\wedge\;\bigwedge_{j}^{i_{j}=1}(v_{j}\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; V_{j})\;\wedge\; \bigwedge_{r}^{i_{r}=\partial}(v_{r}\in V_{r}))\} \end{align} (5)where $$\widehat{\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta }: (X)^{n}\longrightarrow (X)$$ and $$\widehat{\ominus }^{\partial }_{\delta }:(Y)^{n}\longrightarrow (Y)$$. Remark 3.2 The observant reader will notice that $$\widehat{\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta }$$ is the composition of a classical (but two-sorted) image set-operator generated by the relation $$R_{\delta }$$ with the Galois map $$\phi$$ where $$:\prod _{j=1}^{j=n} (Z^{i_{j}})\longrightarrow (X)$$, with $$(Z^{i_{j}})= (X)$$, if $$i_{j}=1$$ and ℘(Y ), when $$i_{j}=\partial$$. A similar observation applies for $$\widehat{\ominus }_{\delta^{\prime}}$$ (with a corresponding two-sorted diamond operator ). Lemma 3.3 Let $${\mathcal G}_{\psi }(X)^{i_{j}}$$ designate $${\mathcal G}_{\psi }(X)$$ if $$i_{j}=1$$, $${\mathcal G}_{\psi }(X)^{\partial }$$ otherwise. Similarly, $${\mathcal G}_{\phi }(Y)^{i_{j}}$$ designates $${\mathcal G}_{\phi }(Y)$$ if $$i_{j}=1$$, $${\mathcal G}_{\phi }(Y)^{\partial }$$ otherwise. If $$\delta \in \tau _{1}$$ and $$U_{j}$$ is a closed element of $${\mathcal G}_{\psi }(X)^{i_{j}}$$, for each j = 1, …, n, then $$\widehat{\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta }(U_{1},\ldots ,U_{n})$$ is a closed element of $${\mathcal G}_{\psi }(X)$$. Similarly, if $$\delta \in \tau _{\partial }$$ and for each r = 1, …, n, $$V_{r}$$ is an open element of $${\mathcal G}_{\phi }(Y)^{i_{r}}$$, then $$\widehat{\ominus }^{\partial }_{\delta }(V_{1},\ldots ,V_{n})\in{\mathcal G}_{\kappa }(Y)$$, hence an open element of $${\mathcal G}_{\phi }(Y)^{\partial }$$. Proof. Recall first (Remark 2.4) that the closed elements of the product $${\mathcal G}_{\psi }(X)^{i_{1}}\times \cdots \times{\mathcal G}_{\psi }(X)^{i_{n}}$$ are tuples of the form $$(\ldots ,\underbrace{\Gamma u_{j}}_{i_{j}=1},\ldots ,\underbrace{{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}\{u_{r}\}}_{i_{r}=\partial },\ldots )$$, with $$u_{j}\in X$$, when $$i_{j}=1$$ and $$u_{r}\in Y$$, when $$i_{r}=\partial$$. For the case $$\delta \in \tau _{1}$$ we verify that $$\widehat{\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta }(\ldots ,\underbrace{\Gamma u_{j}}_{i_{j}=1},\ldots ,\underbrace{{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}\{u_{r}\}}_{i_{r}=\partial },\ldots )=R_{\delta } u_{1}\cdots u_{n}$$. Given definitions, given also that $${ }^{{\mathop{=}^{\kern-5pt\shortmid}}}\{u\}\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; v$$ iff $$v\in \Gamma u$$ iff u ≤ v, the claim reduces to showing that $$xR_{\delta } u_{1}\cdots u_{n}$$ iff $$\exists u^{\prime }_{1}\cdots u^{\prime }_{n}\;(xR_{\delta } u^{\prime }_{1}\cdots u^{\prime }_{n}\;\wedge \;\bigwedge _{i_{j}=1}(u_{j}\leq u^{\prime }_{j})\;\wedge \;\bigwedge _{i_{r}=\partial }(u_{r}\leq u^{\prime }_{r}))$$ iff $$\exists u^{\prime }_{1}\cdots u^{\prime }_{n}\;(xR_{\delta } u^{\prime }_{1}\cdots u^{\prime }_{n}\;\wedge \;\bigwedge _{s}(u_{s}\leq u^{\prime }_{s}))$$. Left to right is obvious and the converse is immediate by the monotonicity properties (R2) of $$R_{\delta }$$. Then $$\widehat{\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta }(\ldots ,\underbrace{\Gamma u_{j}}_{i_{j}=1},\ldots ,\underbrace{{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}\{u_{r}\}}_{i_{r}=\partial },\ldots )=R_{\delta } u_{1}\cdots u_{n}\in{\mathcal G}_{\kappa }(X)$$, by condition (R3) on frames. The argument showing that $$\widehat{\ominus }^{\partial }_{\delta }(\ldots ,\underbrace{\{u_{j}\}{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}}_{i_{j}=1},\ldots ,\underbrace{\Gamma u_{r}}_{i_{r}=\partial },\ldots )=S^{\partial }_{\delta } u_{1}\cdots u_{n}$$ is similar, now using the condition (R4) on frames that $$S^{\partial }_{\delta } u_{1}\cdots u_{n}$$ is a closed element of $${\mathcal G}_{\phi }(Y)$$ (hence an open element of $${\mathcal G}_{\phi }(Y)^{\partial }$$). Corollary 3.4 The generalized image operators $$\widehat{\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta }/\widehat{\ominus }^{\partial }_{\delta }$$ restrict to maps on clopen elements. Proof. Immediate, by Lemma 3.3 and by the restriction in the definition of frames that $$R_{\delta } u_{1}\cdots u_{n}$$ and $$S^{\partial }_{\delta } u_{1}\cdots u_{n}$$ are clopen elements when all the $$u_{i}$$ are clopen. Given the set-operators $$\widehat{\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta }$$ ($$\delta \in \tau _{1}$$) and $$\widehat{\ominus }^{\partial }_{\delta ^{\prime }}$$ ($${\delta ^{\prime }}\in \tau _{\partial }$$), generated by the frame relations $$R_{\delta },S^{\partial }_{\delta ^{\prime }}$$, we now define operators $${\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta },\ominus _{\delta ^{\prime }}$$ on stable sets and their dual operators $${\bigcirc}{\kern-6.4pt\mid}\ \,^{\partial }_{\delta },\ominus ^{\partial }_{\delta ^{\prime }}$$ on co-stable sets, as indicated in the following figure. Definition 3.5 Let $${\bigcirc}{\kern-6.4pt\mid}\ \,^{\partial }_{\delta },\ominus ^{\partial }_{\delta ^{\prime }}:(Y)^{n}\longrightarrow (Y)$$, $$\ominus _{\delta ^{\prime }},{\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta }:(X)^{n}\longrightarrow (X)$$, for $$\delta \in \tau _{1},\;\delta ^{\prime }\in \tau _{\partial }$$, be defined by $${\bigcirc}{\kern-6.4pt\mid}\ \,^{\partial }_{\delta }(V_{1},\ldots ,V_{n}) \;= \;\phi \widehat{\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta }(\psi V_{1},\ldots ,\psi V_{n})$$, for $$V_{j}\subseteq Y$$ $$\ominus _{\delta ^{\prime }}(U_{1},\ldots ,U_{n})=\psi \widehat{\ominus}^{\partial }_{\delta }(\phi U_{1},\ldots ,\phi U_{n})$$, for $$U_{j}\subseteq X$$ $${\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta }=\psi \phi \widehat{\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta }$$ and $$\ominus ^{\partial }_{\delta ^{\prime }}=\phi \psi \widehat{\ominus }_{\delta ^{\prime }}$$. Let also the relations $$R^{\partial }_{\delta }$$ and $$S_{\delta ^{\prime }}$$ be defined be setting   \begin{align} {\hskip7pt}R^{\partial}_{\delta}\subseteq Y\times (X^{i_{1}}\times\cdots\times X^{i_{n}})\ \ \textrm{where}\ \ X^{i_{j}}\;=\;\begin{cases} X & \textrm{if}\ \ i_{j}=1\\ Y & \textrm{if}\ \ i_{j}=\partial \end{cases} \end{align} (6)  \begin{align} {\hskip-85pt}yR^{\partial}_{\delta} u_{1}\cdots u_{n}\equiv&\, \forall x\in X(xR_{\delta} u_{1}\cdots u_{n}\longrightarrow x\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; y)\nonumber\\ \equiv&\, R_{\delta} u_{1}\cdots u_{n}\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; y,\ \textrm{i.e.}\ R^{\partial}_{\delta} u_{1}\cdots u_{n}\;=\;(R_{\delta} u_{1}\cdots u_{n}){{}^{{\mathop{=}^{\kern-5pt\shortmid}}}} \end{align} (7)  \begin{align} {\hskip-10pt}S_{\delta^{\prime}}\subseteq X\times (X^{i_{1}}\times\cdots\times X^{i_{n}})\ \textrm{where}\ X^{i_{j}}=\begin{cases} X & \textrm{if}\ i_{j}=1\\ Y & \textrm{if}\ i_{j}=\partial \end{cases} \end{align} (8)  \begin{align} &xS_{\delta^{\prime}} v_{1}\cdots v_{n}\ \textrm{iff}\quad\ \forall y\in Y\;(yS^{\partial}_{\delta^{\prime}} v_{1}\cdots v_{n}\longrightarrow x\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; y)\ \textrm{iff}\ x\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; S^{\partial}_{\delta^{\prime}} v_{1}\cdots v_{n},\nonumber\\ &\quad\textrm{i.e.}\ S_{\delta^{\prime}} v_{1}\cdots v_{n}\;=\;{}^{{\mathop{=}^{\kern-5pt\shortmid}}}(S^{\partial}_{\delta^{\prime}} v_{1}\cdots v_{n}). \end{align} (9) Lemma 3.6 The following hold, where $$\delta \in \tau _{1},\;\delta ^{\prime }\in \tau _{\partial }$$, $$A_{i}\in{\mathcal G}_{\psi }(X)$$ and $$B_{j}\in{\mathcal G}_{\phi }(Y)$$ 1) $${\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta }(A_{1},\ldots ,A_{n})=\psi \phi (\{x\in X\;|\; \exists u_{1}\cdots u_{n}\;(xR_{\delta } u_{1}\cdots u_{n}\;\wedge \;\bigwedge _{j}^{i_{j}=1}(u_{j}\in A_{j})\;\wedge \;\bigwedge _{r}^{i_{r}=\partial }(A_{r}\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; u_{r}))\})$$$$=\psi \phi \widehat{\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta }(A_{1},\ldots ,A_{n})=\psi {\bigcirc}{\kern-6.4pt\mid}\ \,^{\partial }_{\delta }(\phi A_{1},\ldots ,\phi A_{n})$$ 2) $${\bigcirc}{\kern-6.4pt\mid}\ \,^{\partial }_{\delta }(B_{1},\ldots ,B_{n}) = \{y\in Y\;|\;\forall u_{1}\cdots u_{n}((\bigwedge _{j}^{i_{j}=1}(u_{j}\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; B_{j})\;\wedge \;\bigwedge _{r}^{i_{r}=\partial }(u_{r}\in B_{r}))\;\longrightarrow yR^{\partial }_{\delta } u_{1}\cdots u_{n})\}$$ 3) $$\ominus _{\delta ^{\prime }}(A_{1},\ldots ,A_{n})=\{x\in X\;|\; \forall v_{1}\cdots v_{n}\;((\bigwedge _{j}^{i_{j}=1}(v_{j}\in A_{j})\;\wedge \;\bigwedge _{r}^{i_{r}=\partial }(A_{r}\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; v_{r})\;\longrightarrow \;xS_{\delta ^{\prime }} v_{1}\cdots v_{n}))\}$$ 4) $$\ominus ^{\partial }_{\delta ^{\prime }}(B_{1},\ldots ,B_{n})=\phi \psi (\{y\in Y\;|\; \exists v_{1}\cdots v_{n}\;(yS^{\partial }_{\delta ^{\prime }} v_{1}\cdots v_{n}\;\wedge \;\bigwedge _{j}^{i_{j}=1}(v_{j}\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; V_{j})\;\wedge \; \bigwedge _{r}^{i_{r}=\partial }(v_{r}\in V_{r})\})$$$$=\phi \psi \widehat{\ominus }^{\partial }_{\delta ^{\prime }}(B_{1},\ldots ,B_{n})=\phi \ominus _{\delta ^{\prime }}(\psi B_{1},\ldots ,\psi B_{n})$$ where $$R^{\partial }_{\delta }$$ and $$S_{\delta ^{\prime }}$$ are defined in Definition 3.5. Proof. 1) and 4) are immediate, stated just for explicitness. For 2) we calculate that, for $$B_{i}\in{\mathcal G}_{\phi }(Y)$$, i = 1, …, n,   \begin{align*} &\phi\widehat{\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta}(\psi B_{1},\ldots,\psi B_{n}) \\ &\quad = \{x\in X\;|\; \exists u_{1}\cdots u_{n}\;(xR_{\delta} u_{1}\cdots u_{n}\;\wedge\;\bigwedge_{j}^{i_{j}=1}(u_{j}\in \psi B_{j})\;\wedge\;\bigwedge_{r}^{i_{r}=\partial}(\psi B_{r}\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; u_{r}))\}{^{{\mathop{=}^{\kern-5pt\shortmid}}}}\\ &\quad = \{x\in X\;|\; \exists u_{1}\cdots u_{n}(xR_{\delta} u_{1}\cdots u_{n}\;\wedge\;\bigwedge_{j}^{i_{j}=1}(u_{j}\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; B_{j})\;\wedge\;\bigwedge_{r}^{i_{r}=\partial}(u_{r}\in B_{r}))\}{^{{\mathop{=}^{\kern-5pt\shortmid}}}}\\ &\quad = \{y\in Y\;|\;\forall x\in X((\exists u_{1}\cdots u_{n}(xR_{\delta} u_{1}\cdots u_{n}\;\wedge\bigwedge_{j}^{i_{j}=1}(u_{j}\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; B_{j})\;\wedge\;\bigwedge_{r}^{i_{r}=\partial}(u_{r}\in B_{r}))).\!\!\longrightarrow x\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; y)\}\\ &\quad = \{y\in Y\;|\;\forall x\in X\forall u_{1}\cdots u_{n}(((xR_{\delta} u_{1}\cdots u_{n}\;\wedge\;\bigwedge_{j}^{i_{j}=1}(u_{j}\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; B_{j})\;\wedge\;\bigwedge_{r}^{i_{r}=\partial}(u_{r}\in B_{r}))).\!\!\longrightarrow x\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; y)\}\\ &\quad = \{\,y\in Y\;|\;\forall x\in X\forall u_{1}\cdots u_{n}((\bigwedge_{j}^{i_{j}=1}(u_{j}\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; B_{j})\;\wedge\;\bigwedge_{r}^{i_{r}=\partial}(u_{r}\in B_{r})).\!\longrightarrow(xR_{\delta} u_{1}\cdots u_{n}\longrightarrow x\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; y)\}\\ &\quad = \{\,y\in Y\;|\;\forall u_{1}\cdots u_{n}((\bigwedge_{j}^{i_{j}=1}(u_{j}\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; B_{j})\;\wedge\;\bigwedge_{r}^{i_{r}=\partial}(u_{r}\in B_{r})).\!\longrightarrow\forall x\in X(xR_{\delta} u_{1}\cdots u_{n}\longrightarrow x\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; y)\}\\ &\quad =\{\,y\in Y\;|\;\forall u_{1}\cdots u_{n}((\bigwedge_{j}^{i_{j}=1}(u_{j}\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; B_{j})\;\wedge\;\bigwedge_{r}^{i_{r}=\partial}(u_{r}\in B_{r})).\!\longrightarrow(R_{\delta} u_{1}\cdots u_{n}\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}} \;y)\}\\ &\quad =\{\,y\in Y\;|\;\forall u_{1}\cdots u_{n}((\bigwedge_{j}^{i_{j}=1}(u_{j}\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; B_{j})\;\wedge\;\bigwedge_{r}^{i_{r}=\partial}(u_{r}\in B_{r})).\longrightarrow yR^{\partial}_{\delta} u_{1}\cdots u_{n}\}. \end{align*}2) and 3) are completely similar and we leave details for 3) to the reader. Corollary 3.7 For $$\delta \in \tau _{1}, A_{i}\in{\mathcal G}_{\psi }(X)$$, $${\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta }(A_{1},\ldots ,A_{n})$$ is the join of the $${\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta }(\ldots ,\underbrace{\Gamma u_{j}}_{i_{j}=1},\ldots ,\underbrace{{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}\{u_{r}\}}_{i_{r}=\partial },\ldots )$$ such that for $$i_{j}=1$$, $$u_{j}$$ is in $$A_{j}$$ and for $$i_{r}=\partial$$, $$u_{r}$$ is in $$\phi (A_{r})$$, i.e.   $${\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta}(A_{1},\ldots,A_{n})=\bigvee^{i_{j}=1,u_{j}\in A_{j}}_{i_{r}=\partial,A_{r}\,{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\, u_{r}}R_{\delta} u_{1}\cdots u_{n}= \bigvee^{i_{j}=1,u_{j}\in A_{j}}_{i_{r}=\partial,A_{r}\,{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\, u_{r}}{\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta}(\ldots,\underbrace{\Gamma u_{j}}_{i_{j}=1},\ldots,\underbrace{{}^{{\mathop{=}^{\kern-5pt\shortmid}}}\{u_{r}\}}_{i_{r}=\partial},\ldots).$$Similarly, for $$\delta ^{\prime }\in \tau _{\partial }, B_{i}\in{\mathcal G}_{\phi }(Y)$$  $$\ominus^{\partial}_{\delta^{\prime}}(B_{1},\ldots,B_{n})=\bigvee^{i_{j}=1,v_{j}\,{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\, B_{j}}_{i_{r}=\partial,v_{r}\in B_{r}}S^{\partial}_{\delta^{\prime}}v_{1}\cdots v_{n}=\bigvee^{i_{j}=1,v_{j}\,{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\, B_{j}}_{i_{r}=\partial,v_{r}\in B_{r}}\ominus^{\partial}_{\delta^{\prime}}(\ldots,\underbrace{\{v_{j}\}{}^{{\mathop{=}^{\kern-5pt\shortmid}}}}_{i_{j}=1},\ldots,\underbrace{\Gamma v_{r}}_{i_{r}=\partial},\ldots).$$ Proof.   \begin{align*} {\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta}(A_{1},\ldots,A_{n})&=\psi\phi\bigcup\nolimits_{i_{r}=\partial,A_{r}\,{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\, u_{r}}^{i_{j}=1,u_{j}\in A_{j}}R_{\delta} u_{1}\cdots u_{n} &\textrm{by definition}\\ & =\bigvee\nolimits^{i_{j}=1,u_{j}\in A_{j}}_{i_{r}=\partial,A_{r}\,{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\, u_{r}}\psi\phi R_{\delta} u_{1}\cdots u_{n}&\\ &=\bigvee\nolimits_{i_{r}=\partial,A_{r}\,{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\, u_{r}}^{i_{j}=1,u_{j}\in A_{j}}R_{\delta} u_{1}\cdots u_{n}&R_{\delta} u_{1}\cdots u_{n}\ \textrm{is stable}\\ &=\bigvee\nolimits_{i_{r}=\partial,A_{r}\,{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\, u_{r}}^{i_{j}=1,u_{j}\in A_{j}}\widehat{\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta}(\ldots,\underbrace{\Gamma u_{j}}_{i_{j}=1},\ldots,\underbrace{{}^{{\mathop{=}^{\kern-5pt\shortmid}}}\{u_{r}\}}_{i_{r}=\partial},\ldots)&\textrm{Lemma 3.3}\\ &=\bigvee\nolimits_{i_{r}=\partial,A_{r}\,{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\, u_{r}}^{i_{j}=1,u_{j}\in A_{j}}{\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta}(\ldots,\underbrace{\Gamma u_{j}}_{i_{j}=1},\ldots,\underbrace{{}^{{\mathop{=}^{\kern-5pt\shortmid}}}\{u_{r}\}}_{i_{r}=\partial},\ldots)\!&\!\!\!\textrm{definition of}\ \, {\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta}, \textrm{Lemma 3.3}. \end{align*}The proof for $$\ominus ^{\partial }_{\delta ^{\prime }}$$ is completely similar, left to the reader. The following is then an immediate consequence, tightening the result of Corollary 2. Corollary 3.8 The operators $${\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta },{\bigcirc}{\kern-6.4pt\mid}\ \,^{\partial }_{\delta }$$ and $$\ominus _{\delta ^{\prime }},\ominus ^{\partial }_{\delta ^{\prime }}$$, where $$\delta \in \tau _{1}, \delta ^{\prime }\in \tau _{\partial }$$ restrict to maps on clopen elements of $${\mathcal G}_{\psi }(X)$$ and of $${\mathcal G}_{\phi }(Y)$$, accordingly. As the reader can easily verify, each of $${\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta },\ominus _{\delta ^{\prime }}$$ has the monotonicity properties corresponding to the distribution types $$\delta ,\delta ^{\prime }$$. However, additional axioms must be assumed to ensure that the operators are normal, of distribution type $$\delta ,\delta ^{\prime }$$, respectively. We add appropriate axioms in our definition of $$\tau$$-frames, in Section 3.2. 3.2 $$\tau$$-Frames Let $$\tau =(\tau _{1},\tau _{\partial })$$ be a similarity type with distribution types $$\delta \in \tau _{1}$$ of output type 1 and $$\delta ^{\prime }$$ in $$\tau _{\partial }$$ of output type ∂. Definition 3.9 ($$\tau$$-Frames) A Kripke–Galois 2-sorted $$\tau$$-frame $$\mathfrak{F}=(X,(R_{\delta },R^{\partial }_{\delta })_{\delta \in \tau _{1}},{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}} ,Y,(S_{\delta },S^{\partial }_{\delta })_{\delta \in \tau _{\partial }})$$ is a structure where (X, $${\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}$$, Y) satisfies axioms (F1–F5) for lattice frames, presented in Section 2.2, and the relations $$R_{\delta }$$ with $$\delta \in \tau _{1}$$, $$S^{\partial }_{\delta }$$ with $$\delta \in \tau _{\partial }$$ satisfy axioms (R1–R4), presented in Section 3.1, as well as axioms R5–R8 stated below: (R5) $$R^{\partial }_{\delta }\subseteq Y\times (X^{i_{1}}\times \cdots \times X^{i_{n}})\ \textrm{with}\ X^{i_{j}}\;=\;\left\{\begin{smallmatrix} X & \textrm{if}\ i_{j}=1\\ Y & \textrm{if}\ i_{j}=\partial \end{smallmatrix}\right.$$ and $$\\\forall (u_{1},\ldots , u_{n})\in (X^{i_{1}}\times \cdots \times X^{i_{n}})\;R^{\partial }_{\delta } u_{1}\cdots u_{n}\;=\;(R_{\delta } u_{1}\cdots u_{n}){ }^{{\mathop{=}^{\kern-5pt\shortmid}}}$$ (R6) $$S_{\delta ^{\prime }}\subseteq X\times (X^{i_{1}}\times \cdots \times X^{i_{n}})\ \textrm{with}\ X^{i_{j}}=\left\{\begin{smallmatrix} X & \textrm{if}\ i_{j}=1\\ Y & \textrm{if}\ i_{j}=\partial \end{smallmatrix}\right.$$ and $$\\\forall (v_{1},\ldots ,v_{n})\in (X^{i_{1}}\times \cdots \times X^{i_{n}})\;S_{\delta ^{\prime }} v_{1}\cdots v_{n}\;=\;{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}(S^{\partial }_{\delta ^{\prime }} v_{1}\cdots v_{n})$$ (R7) For all x ∈ X and all $$u_{s},u^{\prime }_{s}\in X^{i_{s}}$$, for all j, s = 1, …, n,   $$xR_{\delta} u_{1}\cdots (u_{j}\cap u^{\prime}_{j})\cdots u_{n}\;\longrightarrow\;\forall y\in Y\;(yR^{\partial}_{\delta} u_{1}\cdots u_{j}\cdots u_{n}\;\wedge\;yR^{\partial}_{\delta} u_{1}\cdots u^{\prime}_{j}\cdots u_{n}\;\longrightarrow\;x\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; y)$$ (R8) For all y ∈ Y and all $$u_{s},u^{\prime }_{s}\in X^{i_{s}}$$, for r, s = 1, …, n,   $$yS^{\partial}_{\delta} u_{1}\cdots (u_{r}\cap u^{\prime}_{r})\cdots u_{n}\;\longrightarrow\;\forall x\in X\;(yS_{\delta} u_{1}\cdots u_{r}\cdots u_{n}\;\wedge\;yS_{\delta} u_{1}\cdots u^{\prime}_{r}\cdots u_{n}\;\longrightarrow\;x\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; y)$$ A Kripke–Galois 2-sorted $$\tau$$-frame will be referred to in the sequel as simply a $$\tau$$-frame. $$\tau$$-frame morphisms are the morphisms $$(\,f,h):(X_{1},{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}} _{1},Y_{1})\longrightarrow (X_{2},{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}} _{2},Y_{2})$$ of their underlying lattice frames (Definition 2.6) where, in addition, $$f^{\ast }=f^{-1},h^{\ast }=h^{-1}$$ are homomorphisms of the respective $$\tau$$-algebras of clopens. We have opted to define $$\tau$$-frames with a pair of relations $$(R_{\delta }, R^{\partial }_{\delta })$$, $$(S_{\delta ^{\prime }},S^{\partial }_{\delta ^{\prime }})$$, for $$\delta \in \tau _{1},\delta ^{\prime }\in \tau _{\partial }$$ so as to be consistent with our definition of Kripke–Galois frames in [14, 16], hence we added axioms (R5, R6) to $$\tau$$-frames. Axioms (R7–R8) are included to ensure that the operators generated by the relations are normal in the sense of the next Proposition. Proposition 3.10 The operator $${\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta }:\prod _{j=1}^{j=n}{\mathcal G}_{\kappa }({\mathcal G}_{\psi }(X)^{i_{j}})\longrightarrow{\mathcal G}_{\kappa }(X)$$ is a normal operator of distribution type $$\delta =(i_{1},\ldots ,i_{n};1)\in \tau _{1}$$, where $${\mathcal G}_{\psi }(X)^{i_{j}}={\mathcal G}_{\psi }(X)$$ when $$i_{j}=1$$ and $${\mathcal G}_{\psi }(X)^{\partial }$$ when $$i_{j}=\partial$$ and $${\mathcal G}_{\kappa }({\mathcal G}_{\psi }(X)^{i_{j}})={\mathcal G}_{\kappa }(X)$$ or it is $${\mathcal G}_{o}(X)$$, respectively. Similarly for $$\ominus _{\delta ^{\prime }}, \delta ^{\prime }\in \tau _{\partial }$$. Proof. We separate cases. Case $$i_{j}=1=i_{n+1}$$: We need to show distribution of $$\ {\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta }$$ over joins at the j-th position, where $$X^{i_{j}}=X$$ and then $$u_{j},u^{\prime }_{j}\in X$$. It suffices to establish that   $${\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta}(\ldots,\Gamma u_{j}\vee\Gamma u^{\prime}_{j},\ldots)\subseteq{\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta}(\ldots,\Gamma u_{j},\ldots)\vee{\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta}(\ldots,\Gamma u^{\prime}_{j},\ldots)$$since the other direction follows from monotonicity. Closed elements form a lattice and this induces a lattice structure on each of X, Y, where $$\Gamma u_{j}\vee \Gamma u^{\prime }_{j}=\Gamma (u_{j}\cap u^{\prime }_{j})$$. Hence we need to show that   $${\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta}(\ldots,\Gamma(u_{j}\cap u^{\prime}_{j}),\ldots)\subseteq{\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta}(\ldots,\Gamma u_{j},\ldots)\vee{\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta}(\ldots,\Gamma u^{\prime}_{j},\ldots).$$By Lemma 3.3, definition of $$\,{\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta }$$ and by the frame axiom (R3), the above requirement is equivalent to the following:   \begin{align*} R_{\delta} u_{1}\cdots (u_{j}\cap u^{\prime}_{j})\cdots u_{n} &\subseteq R_{\delta} u_{1}\cdots u_{j}\cdots u_{n}\;\vee\;R_{\delta} u_{1}\cdots u^{\prime}_{j}\cdots u_{n} \\ &=\psi\phi(R_{\delta} u_{1}\cdots u_{j}\cdots u_{n}\;\cup\;R_{\delta} u_{1}\cdots u^{\prime}_{j}\cdots u_{n})\\ &=\psi((R_{\delta} u_{1}\cdots u_{j}\cdots u_{n}){}^{{\mathop{=}^{\kern-5pt\shortmid}}}\cap (R_{\delta} u_{1}\cdots u^{\prime}_{j}\cdots u_{n}){^{{\mathop{=}^{\kern-5pt\shortmid}}}})\\ &=\psi(R^{\partial}_{\delta} u_{1}\cdots u_{j}\cdots u_{n}\;\cap\;R^{\partial}_{\delta} u_{1}\cdots u^{\prime}_{j}\cdots u_{n}) \end{align*}where we used axiom (R5). Hence the desired inclusion is equivalent to   \begin{align*} xR_{\delta} u_{1}\cdots (u_{j}\cap u^{\prime}_{j})\cdots u_{n}&\longrightarrow x\; {\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\;(R^{\partial}_{\delta} u_{1}\cdots u_{j}\cdots u_{n}\;\cap\;R^{\partial}_{\delta} u_{1}\cdots u^{\prime}_{j}\cdots u_{n}), \text{i.e.}\\ xR_{\delta} u_{1}\cdots (u_{j}\cap u^{\prime}_{j})\cdots u_{n}&\longrightarrow \forall y\in Y\;(yR^{\partial}_{\delta} u_{1}\cdots u_{j}\cdots u_{n}\;\wedge\;yR^{\partial}_{\delta} u_{1}\cdots u^{\prime}_{j}\cdots u_{n}\;\longrightarrow\;x\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; y) \end{align*}which is precisely axiom (R7). Case$$i_{r}=\partial \neq 1= i_{n+1}$$: We need to show co-distribution of $${\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta }$$ over meets at the r-th position, where $$X^{i_{r}}=Y$$, in other words the following needs to be proven, where $$u_{r},u^{\prime }_{r}\in Y$$.   $${\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta}\left(\ldots,{}^{{\mathop{=}^{\kern-5pt\shortmid}}}\{u_{r}\}\cap{}^{{\mathop{=}^{\kern-5pt\shortmid}}}\{u^{\prime}_{r}\},\ldots\right)={\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta}\left(\ldots,{}^{{\mathop{=}^{\kern-5pt\shortmid}}}\{u_{r}\},\ldots\right)\vee{\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta}\left(\ldots,{}^{{\mathop{=}^{\kern-5pt\shortmid}}} \{u^{\prime}_{r}\},\ldots\right).$$Note that $$\phi ({ }^{{\mathop{=}^{\kern-5pt\shortmid}}}\{u_{r}\}\cap{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}\{u^{\prime }_{r}\})=\phi ({ }^{{\mathop{=}^{\kern-5pt\shortmid}}}\{u_{r}\})\vee \phi ({ }^{{\mathop{=}^{\kern-5pt\shortmid}}}\{u^{\prime }_{r}\})=\Gamma u_{r}\vee \Gamma u^{\prime }_{r}=\Gamma (u_{r}\cap u^{\prime }_{r})$$ and therefore $${ }^{{\mathop{=}^{\kern-5pt\shortmid}}}\{u_{r}\}\cap{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}\{u^{\prime }_{r}\}=\psi (\Gamma (u_{r}\cap u^{\prime }_{r}))={ }^{{\mathop{=}^{\kern-5pt\shortmid}}}\{u_{r}\cap u^{\prime }_{r}\}$$. Given Lemma 3.3, given the definition of $$\,{\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta }$$, given also the frame axiom (R3) and the monotonicity properties of the relation $$R_{\delta }$$ prescribed in the frame axiom (R2), it suffices to show that   $$R_{\delta} u_{1}\cdots\left(u_{r}\cap u^{\prime}_{r}\right)\cdots u_{n}\;\subseteq\; R_{\delta} u_{1}\cdots u_{r}\cdots u_{n}\;\vee\;R_{\delta} u_{1}\cdots u^{\prime}_{r}\cdots u_{n}.$$We have $$R_{\delta } u_{1}\cdots u_{r}\cdots u_{n}\;\vee \;R_{\delta } u_{1}\cdots u^{\prime }_{r}\cdots u_{n}=\psi (R^{\partial }_{\delta } u_{1}\cdots u_{r}\cdots u_{n}\;\cap \;R^{\partial }_{\delta } u_{1}\cdots u^{\prime }_{r}\cdots u_{n})$$, by the same argument as in Case 1 above. The rest follows by the frame axiom (R7). Case$$i_{r}=\partial =i_{n+1}$$ Axiom (R8) implies, by an analogous argument to the case $$i_{j}=1=i_{n+1}$$, that the dual operator $$\ominus ^{\partial }_{\delta }$$ distributes over joins in $${\mathcal G}_{\kappa }(Y)$$ at the r-th position. Indeed, axiom (R8) replaces $$R_{\delta }$$ with $$S^{\partial }_{\delta }$$ and interchanges the roles of X, Y in the statement of the condition. Then use duality of the operators $$\ominus _{\delta },\ominus ^{\partial }_{\delta }$$ to derive the desired conclusion. Case$$i_{r}=1\neq \partial =i_{n+1}$$ As in the previous case, axiom (R8) implies, by an argument analogous to the case $$i_{j}=\partial \neq 1=i_{n+1}$$, that $$\ominus ^{\partial }_{\delta }$$ co-distributes over intersections, turning them to joins in $${\mathcal G}_{\kappa }(Y)$$. Then use again the duality of the operators $$\ominus _{\delta },\ominus ^{\partial }_{\delta }$$ to derive the desired conclusion. By Corollary 3.8, the lattice of clopen elements is closed under the operators. Hence we conclude: Corollary 3.11 Each of $$\,{\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta },\ominus _{\delta ^{\prime }}$$ is a normal operator of $${\mathcal G}_{\kappa o}(X)$$ and similarly for the dual operators $${\bigcirc}{\kern-6.4pt\mid}\ \,^{\partial }_{\delta },\ominus ^{\partial }_{\delta ^{\prime }}$$ and $${\mathcal G}_{\kappa o}(Y)$$. The class of $$\tau$$-frames specified above is too large to allow for proving duality with normal lattice expansions. For duality purposes, we distinguish a subclass that satisfies, in addition, axioms (R9, R10) stated below. (R9) For each $$\delta \in \tau _{1}$$ and all $$(u_{1},\ldots ,u_{n})\in X^{i_{1}}\times \cdots \times X^{i_{n}}$$ and z ∈ X, if for all clopen elements $$u^{s}_{*}$$ below $$u_{s}$$, for each s = 1, …, n, there exist $$(v_{1},\ldots ,v_{n})\in X^{i_{1}}\times \cdots \times X^{i_{n}}$$ such that $$zR_{\delta } v_{1}\cdots v_{n}$$ and $$u^{s}_{*}\leq v_{s}$$, for each s, then there exist $$(v^{\prime }_{1},\ldots ,v^{\prime }_{n})\in X^{i_{1}}\times \cdots \times X^{i_{n}}$$ such that $$zR_{\delta } v^{\prime }_{1}\cdots v^{\prime }_{n}$$ and $$u_{s}\leq v^{\prime }_{s}$$, for each s. (R10) For each $$\delta ^{\prime }\in \tau _{\partial }$$ and all $$(u_{1},\ldots ,u_{n})\in X^{i_{1}}\times \cdots \times X^{i_{n}}$$ and y ∈ Y, if for all clopen elements $$u^{s}_{*}$$ below $$u_{s}$$, for each s = 1, …, n, there exist $$(v_{1},\ldots ,v_{n})\in X^{i_{1}}\times \cdots \times X^{i_{n}}$$ such that $$yS^{\partial }_{\delta ^{\prime }} v_{1}\cdots v_{n}$$ and $$u^{s}_{*}\leq v_{s}$$, for each s, then there exist $$(v^{\prime }_{1},\ldots ,v^{\prime }_{n})\in X^{i_{1}}\times \cdots \times X^{i_{n}}$$ such that $$yS^{\partial }_{\delta ^{\prime }} v^{\prime }_{1}\cdots v^{\prime }_{n}$$ and $$u_{s}\leq v^{\prime }_{s}$$, for each s. Remark 3.12 ($$\tau$$-Frames) Henceforth, by a $$\tau$$-frame we mean a frame also satisfying axioms (R9, R10). Proposition 3.13 In a $$\tau$$-frame (satisfying, in addition, axioms (R9, R10), by the previous Remark) the operators $${\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta },\ominus _{\delta ^{\prime }}$$ are completely normal in $${\mathcal G}_{\psi }(X)$$, with distribution types $$\delta$$ and $$\delta ^{\prime }$$, respectively. Their duals $${\bigcirc}{\kern-6.4pt\mid}\ \,^{\partial }_{\delta },\ominus ^{\partial }_{\delta ^{\prime }}$$ are completely normal operators in $${\mathcal G}_{\phi }(Y)$$ with distribution types the duals $$\overline{\delta }, \overline{\delta }^{\prime }$$, respectively. Proof. Recall from Lemma 2.5 that $$(\imath ,{\mathcal G}_{\psi }(X))$$ is a canonical extension of $${\mathcal G}_{\kappa o}(X)$$. Given Corollary 3.7 and the fact that $$\sigma$$-extensions on stable sets are defined using join-density of closed elements, the claim of complete normality of the operators follows if we can show that their restriction to closed elements is the $$\sigma$$-extension of their restriction to clopens. Axioms (R9, R10) are assumed in order to enforce that. The claim to be verified for the case $$\delta \in \tau _{1}$$ is the following, assuming for simplicity of the argument that we designate clopen elements by placing a low index *, as in $$x^{\,j}_{*}$$, $$y^{\,r}_{*}$$,   \begin{align*} &{\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta}(\ldots,\underbrace{\Gamma u_{j}}_{i_{j}=1},\ldots,\underbrace{{}^{{\mathop{=}^{\kern-5pt\shortmid}}}\{u_{r}\}}_{i_{r}=\partial},\ldots)\\ &\quad= \bigcap_{i_{j}=1,u^{j}_{*}\in{\mathcal G}_{\kappa o}(X)}^{i_{r}=\partial,u^{r}_{*}\in{\mathcal G}_{\kappa o}(Y)}\{{\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta}(\ldots,\underbrace{\Gamma u_{*}^{\,j}}_{i_{j}=1},\ldots,\underbrace{{}^{{\mathop{=}^{\kern-5pt\shortmid}}}\{u_{*}^{r}\}}_{i_{r}=\partial},\ldots) \;\Big|\; \bigwedge_{i_{j}=1}(\Gamma u_{j}\subseteq\Gamma u_{*}^{\,j})\wedge\bigwedge_{i_{r}=\partial}({}^{{\mathop{=}^{\kern-5pt\shortmid}}}\{u_{*}^{r}\}\subseteq{}^{{\mathop{=}^{\kern-5pt\shortmid}}}\{u_{r}\}) \} \end{align*}since the expression to the right of equality defines precisely the $$\sigma$$-extension $${\bigcirc}{\kern-6.4pt\mid}\ \,^{\sigma }_{\delta }$$ of the restriction of the operator $${\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta }$$ to clopens. The inclusion left-to-right follows from monotonicity properties, hence the desired inclusion, slightly restated, is the following:   $$\bigcap_{i_{j}=1,u^{j}_{*}\leq u_{j}, u^{\,j}_{*}\in{\mathcal G}_{\kappa o}(X)}^{i_{r}=\partial,u^{r}_{*}\leq u_{r},u^{r}_{*}\in{\mathcal G}_{\kappa o}(Y)}{\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta}(\ldots,\underbrace{\Gamma u_{*}^{j}}_{i_{j}=1},\ldots,\underbrace{{}^{{\mathop{=}^{\kern-5pt\shortmid}}}\{u_{*}^{r}\}}_{i_{r}=\partial},\ldots) \;\;\subseteq \;\;{\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta}(\ldots,\underbrace{\Gamma u_{j}}_{i_{j}=1},\ldots,\underbrace{{}^{{\mathop{=}^{\kern-5pt\shortmid}}}\{u_{*}^{r}\}}_{i_{r}=\partial},\ldots)$$which is equivalent to the claim that for all z ∈ X, if for all clopen elements $$u^{s}_{*}$$ below $$u_{s}$$, for each s = 1, …, n, there exist $$(v_{1},\ldots ,v_{n})\in X^{i_{1}}\times \cdots \times X^{i_{n}}$$ such that $$zR_{\delta } v_{1}\cdots v_{n}$$ and $$u^{s}_{*}\leq v_{s}$$, for each s, then there exist $$(v^{\prime }_{1},\ldots ,v^{\prime }_{n})\in X^{i_{1}}\times \cdots \times X^{i_{n}}$$ such that $$zR_{\delta } v^{\prime }_{1}\cdots v^{\prime }_{n}$$ and $$u_{s}\leq v^{\prime }_{s}$$ for each s = 1, …, n. But this is precisely axiom (R9). For the case $$\delta ^{\prime }\in \tau _{\partial }$$, we merely prove the claim for the dual operator $$\ominus ^{\partial }_{\delta ^{\prime }}$$, by dualising the argument and using axiom (R10). The corresponding claim for $$\ominus _{\delta ^{\prime }}$$ follows from the fact that $$\ominus _{\delta ^{\prime }}$$ is the dual of $$\ominus ^{\partial }_{\delta ^{\prime }}$$. To be precise, the argument outlined above shows that $$\ominus _{\delta ^{\prime }}$$ is the $$\pi$$-extension of its restriction to clopens. However, by normality (Proposition 1) combining with the results of [9] (Lemmas 4.3, 4.4, 4.6) $$\sigma$$ and $$\pi$$-extensions coincide for normal operators. 4 Stone duality for normal lattice expansions In this section we show that the duality of the categories of bounded lattices and ⊥-frames proven in [19] lifts to a duality of the categories of normal lattice expansions of similarity type $$\tau$$ with the corresponding categories of $$\tau$$-frames. Let $$\mathfrak{F}_{\tau }$$ be a $$\tau$$-frame. Its complex algebra $$\mathfrak{F}^{+}_{\tau }=\hat{C}(\mathfrak{F}_{\tau })$$ is the algebra of clopens of $${\mathcal G}_{\psi }(X)$$ (which is a subalgebra of the full complex algebra of stable sets of the frame) and its dual complex algebra is the algebra of clopens of $${\mathcal G}_{\phi }(Y)$$. By Corollary 3.13, the lattice of clopens with the operators $${\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta },\ominus _{\delta ^{\prime }}$$ generated by the relations $$R_{\delta },S_{\delta ^{\prime }}$$, with $$\delta \in \tau _{1},\delta ^{\prime }\in \tau _{\partial }$$, is a normal lattice expansion, hence an object of $$\textbf{L}_{\tau }$$. Conversely, we consider lattice expansions $${\mathcal A}_{\tau }=({\mathcal L},(f_{\delta })_{\delta \in \tau _{1}},(h_{\delta })_{\delta \in \tau _{\partial }})$$ with normal operators $$(f_{\delta })_{\delta \in \tau _{1}}$$ and $$(h_{\delta })_{\delta \in \tau _{\partial }}$$, where $${\mathcal L}=(L,\wedge ,\vee ,0,1)$$ is the underlying bounded lattice and we construct their dual frame $$\hat{F}({\mathcal A}_{\tau })=({\mathcal A}_{\tau })_{+}$$. Let (X, $${\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}$$, Y) be the dual frame of the lattice, after [19] (hence satisfying the frame axioms (F1–F5)), where X, Y are the sets of filters and ideals, respectively of the lattice and $${\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}$$ ⊆ X × Y is defined by x$${\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}$$y iff $$\,x\cap y\neq \emptyset$$. To extend the lattice frame to a $$\tau$$-frame we rely on the definitions of filter/ideal operators we introduced in [12]. Case$$\,\delta \in \tau _{1}$$: For each $$\delta =(i_{1},\ldots ,i_{n};1)\in \tau _{1}$$ and normal lattice operator $$f_{\delta }:{\mathcal L}^{i_{1}}\times \cdots \times{\mathcal L}^{i_{n}}\longrightarrow{\mathcal L}$$ define the relation $$R_{\delta }\subseteq X\times (X^{i_{1}}\times \cdots \times X^{i_{n}})\ \textrm{where}\ X^{i_{j}}=\left\{\begin{smallmatrix}\!\! X & \textrm{if}\ i_{j}=1\\ \!\! Y & \textrm{if}\ i_{j}=\partial \end{smallmatrix}\right.$$ by setting $$xR_{\delta } u_{1}\cdots u_{n}\;\ \textrm{iff}\ \;f^{\flat }_{\delta }(u_{1},\ldots ,u_{n})\leq x$$, where ≤ designates filter inclusion and $$f^{\flat }_{\delta}\!:X^{i_{1}}\times \cdots \times X^{i_{n}}\!\longrightarrow\! X$$, where $$X^{i_{j}}=X$$ if $$i_{j}=1$$ and $$X^{i_{j}}=Y$$ when $$i_{j}=\partial$$, is defined by (10), after [12],   \begin{align} f^{\flat}_{\delta}(u_{1},\ldots,u_{n})=\bigvee\left\{x_{fa_{1}\cdots a_{n}}\;\Big|\; \bigwedge_{j}(a_{j}\in u_{j})\right\} \end{align} (10)where we systematically designate principal filters by $$x_{e}=e\uparrow$$ and principal ideals by $$y_{e}=e\downarrow$$. The relation $$R^{\partial }_{\delta }\subseteq Y\times (X^{i_{1}}\times \cdots \times X^{i_{n}})$$ is defined by condition (7), now taking the form   $$yR^{\partial}_{\delta} u_{1}\cdots u_{n}\ \textrm{iff}\ R_{\delta} u_{1}\cdots u_{n}\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; y\ \textrm{iff}\ f^{\flat}_{\delta}(u_{1},\ldots,u_{n})\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; y.$$ Case$$\,\delta ^{\prime }\in \tau _{\partial }$$: Similarly, for each $$\delta ^{\prime }\in \tau _{\partial }$$ and normal lattice operator $$h_{\delta ^{\prime }}:{\mathcal L}^{i_{1}}\times \cdots \times{\mathcal L}^{i_{n}}\longrightarrow{\mathcal L}^{\partial }$$, define $$S^{\partial }_{\delta ^{\prime }}\subseteq Y\times (X^{i_{1}}\times \cdots \times X^{i_{n}})$$, where $$X^{i_{s}}=X$$ if $$i_{s}=1$$ and $$X^{i_{s}}=Y$$ if $$i_{s}=\partial$$, by setting $$yS^{\partial }_{\delta ^{\prime }}u_{1}\cdots u_{n}$$ iff $$h^{\sharp }_{\delta ^{\prime }} (u_{1},\ldots ,u_{n})\leq y$$, where $$h^{\sharp }_{\delta ^{\prime }}:X^{i_{1}}\times \cdots \times X^{i_{n}}\longrightarrow Y$$, where $$X^{i_{s}}=X$$, when $$i_{s}=1$$ and $$X^{i_{s}}=Y$$ when $$i_{s}=\partial$$ is defined by equation (11), after [12],   \begin{align} h^{\sharp}_{\delta^{\prime}}(u_{1},\ldots,u_{n})=\bigvee\left\{y_{ha_{1}\cdots a_{n}}\;\Big|\; \bigwedge_{j}(a_{j}\in u_{j})\right\} \end{align} (11)where recall that $$y_{e}$$ designates a principal ideal. The definition of the relation $$S_{\delta ^{\prime }}$$ in (9) now takes the form   $$xS_{\delta^{\prime}}u_{1}\cdots u_{n}\ \textrm{iff}\ x\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; S^{\partial}_{\delta^{\prime}}u_{1}\cdots u_{n} \textrm{ iff}\ x\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; h^{\sharp}_{\delta^{\prime}}(u_{1},\ldots,u_{n}).$$ By definition of the canonical relations $$R_{\delta }, S^{\partial }_{\delta ^{\prime }}$$ the frame axiom (R1) holds for the canonical frame. Theorem 4.1 (1) The canonical relations $$R_{\delta }, S^{\partial }_{\delta ^{\prime }}$$ are increasing in the first place and decreasing in every other place (frame axiom (R2)). (2) For any $$(u_{1},\ldots ,u_{n})\in X^{i_{1}}\times \cdots \times X^{i_{n}}$$, $$R_{\delta } u_{1}\cdots u_{n}$$ is a closed element of $${\mathcal G}_{\psi }(X)$$ and $$S^{\partial }_{\delta ^{\prime }}u_{1}\cdots u_{n}$$ is a closed element of $${\mathcal G}_{\phi }(Y)$$. Furthermore, if all the $$u_{i}$$ are clopen points, then so is each of $$R_{\delta } u_{1}\cdots u_{n}, S^{\partial }_{\delta ^{\prime }}u_{1}\cdots u_{n}$$ (frame axioms (R3, R4)). (3) For each $$\delta \in \tau _{1}, \delta ^{\prime }\in \tau _{\partial }$$, $${\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta }=f_{\delta }^{\sigma }$$ and $$\ominus _{\delta ^{\prime }}=h^{\sigma }_{\delta ^{\prime }}$$. (4) The canonical frame is a $$\tau$$-frame, i.e. axioms (F1–F5) and (R1–R10) hold. Proof. For 1), each of $$f^{\flat }_{\delta }, h^{\sharp }_{\delta ^{\prime }}$$ is monotone in each argument place. From this fact and from the definition the monotonicity properties claimed for the canonical relations follow. For 2), it is immediate from the definition of the relations that   \begin{align} R_{\delta} u_{1}\cdots u_{n}=\{x\in X\;|\; xR_{\delta} u_{1}\cdots u_{n}\}=\{x\in X\;|\; f^{\flat}_{\delta}(u_{1},\ldots,u_{n})\leq x\}=\Gamma(\,f^{\flat}_{\delta}(u_{1},\ldots,u_{n})) \end{align} (12)hence $$R_{\delta } u_{1}\cdots u_{n}$$ is a closed element of $${\mathcal G}_{\psi }(X)$$. By [12], Lemma 6.7, $$f^{\flat }_{\delta }$$ preserves principal filters/ideals, hence $$R_{\delta }$$ is clopen, when all the $$u_{i}$$ are clopen points (principal filters, or principal ideals). Similarly,   \begin{align} S^{\partial}_{\delta^{\prime}}u_{1}\cdots u_{n}=\Gamma(h^{\sharp}_{\delta^{\prime}}(u_{1},\ldots,u_{n}))\qquad xS_{\delta^{\prime}}u_{1}\cdots u_{n}\equiv x\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; h^{\sharp}_{\delta^{\prime}}(u_{1},\ldots,u_{n}) \end{align} (13)hence $$S^{\partial }_{\delta ^{\prime }}$$ is a closed element of $${\mathcal G}_{\phi }(Y)$$ and, by [12], Lemma 6.7, $$h^{\sharp }_{\delta ^{\prime }}$$ preserves principal filters/ideals, hence $$S^{\partial }_{\delta ^{\prime }}$$ is clopen, when all the $$u_{i}$$ are clopen points. For 3), by the proof of Lemma 3.3 (instantiated to the canonical frame, already verified to satisfy axioms (F1–F5) and (R1–R4)) and by equation (12), $$\widehat{\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta }(\ldots ,\underbrace{\Gamma u_{j}}_{i_{j}=1},\ldots ,\underbrace{{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}\{u_{r}\}}_{i_{r}=\partial },\ldots )=R_{\delta } u_{1}\cdots u_{n}=\Gamma (\,f^{\flat }_{\delta }(u_{1},\ldots ,u_{n}))$$, hence we obtain $$\psi \phi \widehat{\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta }(\ldots ,\underbrace{\Gamma u_{j}}_{i_{j}=1},\ldots ,\underbrace{{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}\{u_{r}\}}_{i_{r}=\partial },\ldots )={\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta }(\ldots ,\underbrace{\Gamma u_{j}}_{i_{j}=1},\ldots ,\underbrace{{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}\{u_{r}\}}_{i_{r}=\partial },\ldots )=\Gamma (\,f^{\flat }_{\delta }(u_{1},\ldots ,u_{n}))$$. To complete the proof of claim 3), the following lemma is needed. Lemma 4.2 $$\Gamma (f^{\flat }_{\delta }(u_{1},\ldots ,u_{n}))=f^{\sigma }_{\delta }(\ldots ,\underbrace{\Gamma u_{j}}_{i_{j}=1},\ldots ,\underbrace{{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}\{u_{r}\}}_{i_{r}=\partial },\ldots )$$. Proof. We first illustrate the proof for the case of a unary monotone map. We show that if $$\delta (f)=(1;1)$$, then $$f_{\sigma }(\Gamma x)=\Gamma (f^{\flat } x)$$. Recall that the $$\sigma$$ extension $$f^{\sigma }:{\mathcal G}_{\psi }(X)\longrightarrow{\mathcal G}_{\psi }(X)$$ of a monotone map f as in equation (1) is defined by instantiating equation (1) in the dual lattice frame of [19] by setting   \begin{align} f^{\sigma}(\Gamma x)=&\,\bigwedge\left\{\alpha_{X}(\,fa)\;|\; a\in{\mathcal L}, \Gamma x\leq\alpha_{X}(a)\right\}=\bigwedge\{\Gamma x_{fa}\;|\;\Gamma x\subseteq\Gamma x_{a}\}\nonumber\\ =&\,\bigwedge\{\Gamma x_{fa}\;|\; a\in x\}=\Gamma\left(\bigvee\{x_{fa}\;|\; a\in x\}\right). \end{align} (14)By (10), $$\,f^{\flat } x=\bigvee \{x_{fa}\;|\; a\in x\}$$, hence $$f^{\sigma }(\Gamma x)=\Gamma (\,f^{\flat } x)$$, qed. Consider now the general case of a distribution type $$\delta =(i_{1},\ldots ,i_{n};1)$$, of output type 1. Literally, the $$\sigma$$-extension defined in [9] is a map $$f^{\sigma }:{\mathcal G}(Z^{i_{1}})\times \cdots \times{\mathcal G}(Z^{i_{n}})\longrightarrow{\mathcal G}_{\psi }(X)$$, where $${\mathcal G}(Z^{i_{j}})=\left\{\begin{smallmatrix} {\!\!\mathcal G}_{\psi }(X) &\textrm{when}\ i_{j}=1\\ {\!\!\mathcal G}_{\phi }(Y) &\textrm{when}\ i_{j}=\partial \end{smallmatrix}\right.$$, defined on closed elements by $$f^{\sigma }(\Gamma u_{1},\ldots ,\Gamma u_{n})=\bigwedge \{\Gamma x_{fa_{1}\cdots a_{n}}\;|\;\bigwedge _{j}(a_{j}\in u_{j})\}$$. Composing with the dual equivalence $$(\psi ,\phi )$$ at the appropriate argument places we get the equivalent definition of the map $$f^{\sigma }:{\mathcal G}_{\psi }(X)^{i_{1}}\times \cdots \times{\mathcal G}_{\psi }(X)^{i_{n}}\longrightarrow{\mathcal G}_{\psi }(X)$$, where $${\mathcal G}_{\psi }(X)^{i_{j}}=\left\{\begin{smallmatrix} {\!\!\!\!\!\mathcal G}_{\psi }(X) & \textrm{when}\ i_{j}=1\\ {\!\!\mathcal G}_{\psi }(X)^{\partial } & \textrm{when}\ i_{j}=\partial \end{smallmatrix}\right.$$, defined by $$f^{\sigma }(\ldots ,\underbrace{\Gamma u_{j}}_{i_{j}=1},\ldots ,\underbrace{{}{^{{\mathop{=}^{\kern-5pt\shortmid}}}}\{u_{r}\}}_{i_{r}=\partial },\ldots )=\bigwedge \{\Gamma x_{fa_{1}\cdots a_{n}}\;|\;\bigwedge _{j}(a_{j}\in u_{j})\}=\Gamma (\bigvee \{x_{fa_{1}\cdots a_{n}}\;|\;\bigwedge _{j}(a_{j}\in u_{j})\})$$. Hence we obtain $$f^{\sigma }(\ldots ,\underbrace{\Gamma u_{j}}_{i_{j}=1},\ldots ,\underbrace{{}{^{{\mathop{=}^{\kern-5pt\shortmid}}}}\{u_{r}\}}_{i_{r}=\partial },\ldots )=\Gamma (f^{\flat } (u_{1},\ldots , u_{n}))$$. We may then conclude that $$\,{\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta }(\ldots ,\underbrace{\Gamma u_{j}}_{i_{j}=1},\ldots ,\underbrace{{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}\{u_{r}\}}_{i_{r}=\partial },\ldots )=\Gamma (f^{\flat }_{\delta }(u_{1},\ldots ,u_{n}))=f^{\sigma }(\ldots ,\underbrace{\Gamma u_{j}}_{i_{j}=1},\ldots,\underbrace{{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}\{u_{r}\}}_{i_{r}=\partial },\ldots )$$. Use now Corollary 3.7, for the particular case of the canonical frame, where it was shown that   $${\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta}(A_{1},\ldots,A_{n})=\bigvee^{i_{j}=1,u_{j}\in A_{j}}_{i_{r}=\partial,A_{r}{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}} u_{r}}{\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta}(\ldots,\underbrace{\Gamma u_{j}}_{i_{j}=1},\ldots,\underbrace{{}^{{\mathop{=}^{\kern-5pt\shortmid}}}\{u_{r}\}}_{i_{r}=\partial},\ldots)$$together with the fact that the $$\sigma$$-extension of f is defined on all stable sets in [9] using join-density of closed elements as precisely the join displayed above to conclude that $$f^{\sigma }_{\delta }={\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta }$$ on $${\mathcal G}_{\psi }(X)$$. For the reader’s benefit, we display the calculation.   \begin{align*} {\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta}(A_{1},\ldots,A_{n}) &=\psi\phi\widehat{\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta}(A_{1},\ldots,A_{n})\\ &=\psi\phi\left(\{x\in X\;|\;\exists u_{1}\cdots u_{n}\;(xR_{\delta} u_{1}\cdots u_{n}\;\wedge\;\bigwedge_{i_{j}=1}(u_{j}\in A_{j}) \;\wedge\;\bigwedge_{i_{r}=\partial}(A_{r}\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; u_{r})) \}\right)\\ &=\psi\phi\left( \bigcup^{i_{j}=1,u_{j}\in A_{j}}_{i_{r}=\partial,A_{r}\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; u_{r}}\{x\in X\;|\; xR_{\delta} u_{1}\cdots u_{n}\}\right)=\psi\phi\left( \bigcup^{i_{j}=1,u_{j}\in A_{j}}_{i_{r}=\partial,A_{r}\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; u_{r}}\Gamma(\,f^{\flat}_{\delta}(u_{1},\ldots,u_{n}))\right)\\ &=\bigvee^{i_{j}=1,u_{j}\in A_{j}}_{i_{r}=\partial,A_{r}\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; u_{r}}\psi\phi\Gamma\left(\,f^{\flat}_{\delta}(u_{1},\ldots,u_{n})\right)=\bigvee^{i_{j}=1,u_{j}\in A_{j}}_{i_{r}=\partial,A_{r}\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; u_{r}}\Gamma\left(\,f^{\flat}_{\delta}(u_{1},\ldots,u_{n})\right)\\ &=\bigvee^{i_{j}=1,u_{j}\in A_{j}}_{i_{r}=\partial,A_{r}\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; u_{r}}f^{\sigma}_{\delta}(\ldots,\underbrace{\Gamma u_{j}}_{i_{j}=1},\ldots, \underbrace{{}{{}^{{\mathop{=}^{\kern-5pt\shortmid}}}}\{u_{r}\}}_{i_{r}=\partial},\ldots)= f^{\sigma}_{\delta}(A_{1},\ldots,A_{n}). \end{align*}This proves the first subclaim of claim 3). For the second subclaim of 3), that $$\ominus _{\delta ^{\prime }}=h^{\sigma }_{\delta ^{\prime }}$$, we just dualize the argument. Indeed, if $$h_{\delta ^{\prime }}:{\mathcal L}^{i_{1}}\times \cdots \times{\mathcal L}^{i_{n}}\longrightarrow{\mathcal L}^{\partial }$$, let $${\mathcal M}={\mathcal L}^{\partial }$$ so that $$h_{\delta ^{\prime }}:{\mathcal M}^{\overline{i_{1}}}\times \cdots \times{\mathcal M}^{\overline{i_{n}}}\longrightarrow{\mathcal M}$$, where $$\overline{i_{j}}$$ is the dual value, i.e. ∂ if $$i_{j}=1$$ and 1 if $$i_{j}=\partial$$. Then h appears as a map of the dual distribution type $$\overline{\delta ^{\prime }}$$ on $${\mathcal M}$$, therefore of output type 1. Hence the previous argument can be dualized, interchanging $${\mathcal G}_{\phi }(Y)$$ for $${\mathcal G}_{\psi }(X)$$, open for closed etc. Writing $$h^{\sigma ,\partial }_{\delta ^{\prime }}$$ for the dual $$\sigma$$-extension of h and composing with the dual equivalence $$\psi ,\phi$$ we obtain what is called the $$\pi$$-extension of h in [9]. We illustrate this for a unary map below.   \begin{align*} \psi\left(\,f_{\sigma}^{\partial}(\Gamma y)\right) =&\,\psi\left(\bigwedge\{\Gamma y_{fa}\;|\; a\in y\}\right) =\bigvee\{\psi\Gamma y_{fa}\;|\; a\in y\} =\bigvee\{\Gamma x_{fa}\;|\; a\in y\} \\ =&\, \bigvee\{\Gamma x_{fa}\;|\; y_{a}\leq y\} =\bigvee\{\Gamma x_{fa}\;|\; \Gamma y\subseteq \Gamma y_{a}\} =\bigvee\{\Gamma x_{fa}\;|\; \psi\Gamma y_{a}\subseteq \psi\Gamma y\} \\ =&\,\bigvee\{\Gamma x_{fa}\;|\;{}{^{{\mathop{=}^{\kern-5pt\shortmid}}}} \{y_{a}\}\subseteq{}{{}^{{\mathop{=}^{\kern-5pt\shortmid}}}} \{y\}\} =\bigvee\{\Gamma x_{fa}\;|\; \alpha_{Y}(a)\subseteq{}{^{{\mathop{=}^{\kern-5pt\shortmid}}}} \{y\}\} =f_{\pi}({}{^{{\mathop{=}^{\kern-5pt\shortmid}}}}\{y\}) \end{align*}The above argument has shown that $$\ominus _{\delta ^{\prime }}=h^{\pi }_{\delta ^{\prime }}$$. By [9], Lemmas 4.3, 4.4, 4.6, the following hold for a unary monotone map f (1) The $$\sigma$$ and $$\pi$$ extensions $$f^{\sigma }, f^{\pi }$$ of f agree on closed or open elements. (2) If either $$f^{\sigma }$$ preserves all joins, or $$f^{\pi }$$ preserves all meets, then $$f^{\sigma }= f^{\pi }$$. (3) If f preserves binary joins then $$f^{\sigma }$$ preserves all joins and if f preserves binary meets, then $$f^{\pi }$$ preserves all meets. Note that the above hold for any map since e.g. an antitone map from $$\mathcal L$$ to $$\mathcal K$$ is a monotone map from $$\mathcal L$$ to $${\mathcal K}^{\partial }$$ and an n-ary map can be regarded as a unary map whose domain is the product lattice. Therefore, since the operators of interest in this article are assumed to be normal, $$\sigma$$ and $$\pi$$-extensions coincide and this concludes the proof of claim 3). To prove claim 4) we essentially only need to verify the following. Lemma 4.3 If g is a normal lattice operator of some distribution type $$\delta$$, then its $$\sigma$$-extension $$g^{\sigma }$$ is a completely normal operator of the same distribution type. Proof. This can be established in one of two ways. Indeed, we may appeal to [12], Theorem 6.6. Alternatively, we may appeal to [9], Lemma 4.6. Either way, the claim of the Lemma is established. We may now complete the proof of Theorem 4.1, showing that the canonical frame of a lattice expansion is a $$\tau$$-frame. We have already pointed out that the frame axioms (F1–F5) clearly hold for the canonical frame, as do axioms (R1–R4). Axioms (R5, R6) hold trivially since the dual relations are defined so as to satisfy the axioms. The frame axioms (R7, R8) were included in order to prove that the generated operators $${\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta },\ominus _{\delta ^{\prime }}$$ have the requisite distribution properties and they are merely a rephrasing of particular instances of (co)distribution. Hence they hold in the canonical frame, following from the fact that the operators generated by the relations were shown to have the complete (co)distribution properties for the distribution types $$\delta ,{\delta ^{\prime }}$$, respectively. Finally, axioms (R9, R10) are equivalent to the claim that the operators $${\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta },\ominus _{\delta ^{\prime }}$$ are the $$\sigma$$-extensions of their restriction to clopen elements and the proof of the latter was the content of claim 3), proven above. This establishes that the dual frame of a normal lattice expansion is a $$\tau$$-frame and it completes the proof of Theorem 4.1. We have then defined, by the above, the object part of functors $$\hat{C}:\textbf{L}_{\tau }\longrightarrow{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}\textbf{Frm}_{\tau }$$, sending a lattice expansion of similarity type $$\tau$$ to its dual frame, and $$\hat{F}:{}{^{{\mathop{=}^{\kern-5pt\shortmid}}}}\textbf{Frm}_{\tau }\longrightarrow \textbf{L}_{\tau }$$, sending a frame to its dual algebra of clopens. Lemma 4.4 $$\hat{C}:\textbf{L}_{\tau }\longrightarrow{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}\textbf{Frm}_{\tau }$$, sending a lattice expansion of similarity type $$\tau$$ to its dual frame, and $$\hat{F}:{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}\textbf{Frm}_{\tau }\longrightarrow \textbf{L}_{\tau }$$, sending a frame to its dual algebra of clopens extend to contravariant functors. Proof. The proof is an extension of the corresponding proof for the underlying categories of lattices and lattice frames given in [19]. By the results of [19], given a homomorphism $$f:{\mathcal A}\longrightarrow{\mathcal B}$$ of $$\tau$$-algebras we obtain a lattice frame morphism $$\hat{F}(f)=(f_{1},f_{2}):(X_{B},{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}} _{B},Y_{B})=\hat{F}({\mathcal B})\longrightarrow \hat{F}({\mathcal A})=(X_{A},{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}} _{A},Y_{A})$$ where $$f_{1}(x)=f^{-1}[x]$$ and $$f_{2}(y)=f^{-1}[y]$$ such that their inverses $$f_{1}^{\ast }=f_{1}^{-1}:X^{\ast }_{A}\longrightarrow X^{\ast }_{B}$$, $$f_{2}^{\ast }=f_{2}^{-1}:Y^{\ast }_{A}\longrightarrow Y^{\ast }_{B}$$ are lattice homomorphisms (where $$X^{\ast }={\mathcal G}_{\kappa o}(X), Y^{\ast }={\mathcal G}_{\kappa o}(Y)$$) satisfying $$f_{1}^{\ast }(\alpha _{A}(a))=\alpha _{B}(b)$$ iff $$f_{2}^{\ast }(\beta _{A}(a))=\beta _{B}(b)$$ iff f(a) = b, where $$\alpha ,\beta$$ are the (co)representation maps, $$\alpha (e)=\{x\in X\;|\; e\in x\}$$ and $$\beta (e)=\{y\in Y\;|\; e\in y\}$$ (with the corresponding subscripts A, B). For a distribution type $$\delta$$, let $$\oplus _{\delta ,A}:{\mathcal A}^{n}\longrightarrow{\mathcal A}$$ and $$\oplus _{\delta ,B}:{\mathcal B}^{n}\longrightarrow{\mathcal B}$$ be the corresponding operators in $${\mathcal A},{\mathcal B}$$, respectively, and $$\oplus _{\delta ,A}^{\sigma }:(X^{\ast }_{A})^{n}\longrightarrow X^{\ast }_{A}$$, $$\oplus _{\delta ,B}^{\sigma }:(X^{\ast }_{B})^{n}\longrightarrow X^{\ast }_{B}$$ their respective representations (their $$\sigma$$-extensions, restricted to clopen sets). We verify that $$f^{\ast }_{1}(\oplus _{\delta ,A}^{\sigma }(A_{1},\ldots ,A_{n}))=\oplus _{\delta ,B}^{\sigma }(f^{\ast }_{1}(A_{1}),\ldots ,f^{\ast }_{1}(A_{n}))$$. To this purpose, let $$A_{i}=\alpha _{A}(a_{i})$$, so that we have   \begin{array}{lll} f^{\ast}_{1}(\oplus_{\delta,A}^{\sigma}(A_{1},\ldots,A_{n})){\hskip-8pt}&=f^{\ast}_{1}(\oplus_{\delta,A}^{\sigma}(\alpha_{A}(a_{1}),\ldots,\alpha_{A}(a_{n}))) & \textrm{by}\ A_{i}=\alpha_{A}(a_{i})\\ &=f^{\ast}_{1}(\alpha_{A}(\oplus_{\delta,A}(a_{1},\ldots,a_{n}))) & \sigma \text{-extensions represent the operators}\\ &=\alpha_{B}(\,f(\oplus_{\delta,A}(a_{1},\ldots,a_{n})))& \textrm{by definition of}\ f^{\ast}_{1} \ \textrm{and results of}\ (19)\\ &=\alpha_{B}(\oplus_{\delta,B}(\,f(a_{1}),\ldots,f(a_{n}))) & f:{\mathcal A}\longrightarrow{\mathcal B}\ \textrm{is a homomorphism}\\ &=\oplus^{\sigma}_{\delta,B}(\alpha_{B}(\,f(a_{1})),\ldots,\alpha_{B}(\,f(a_{n}))) & \sigma \text{-extensions represent the operators}\\ &=\oplus^{\sigma}_{\delta,B}(\,f^{\ast}_{1}(\alpha_{A}(a_{1})),\ldots,f^{\ast}_{1}(\alpha_{A}(a_{n}))) & \textrm{by definition of}\ f^{\ast}_{1} \ \textrm{and results of}\ (19)\\ &=\oplus^{\sigma}_{\delta,B}(\,f^{\ast}_{1}(A_{1}),\ldots,f^{\ast}_{1}(A_{n})) & \textrm{by}\ A_{i}=\alpha_{A}(a_{i}) \end{array}Hence $$f^{\ast }_{1}$$ is a homomorphism and then $$\hat{F}$$ is fully defined as a contravariant functor from L$$_{\tau }$$ to $${ }^{{\mathop{=}^{\kern-5pt\shortmid}}}\textbf{Frm}_{\tau }$$. By [19], Proposition 2.11, every lattice frame (X, $${\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}$$, Y) is the dual of a lattice $${\mathcal L}$$, i.e. it can be concretely viewed as a frame consisting of the sets X, Y of filters and ideals of $$\mathcal L$$ and where the Galois relation $${\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}$$ is the relation x$${\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}$$y iff $$x\cap y\neq \emptyset$$. Furthermore, by [19], Proposition 2.11 again, every lattice frame morphism $$(\,f_{1},f_{2}):(X_{B},{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}} _{B},Y_{B})\longrightarrow (X_{A},{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}} _{A},Y_{A})$$ arises from a unique lattice homomorphism $$f:{\mathcal A}\longrightarrow{\mathcal B}$$, where f(a) = b iff $$\,f^{\ast }_{1}(\alpha _{A}(a))=\alpha _{B}(b)$$ and then $$f_{1}(x)=f^{-1}[x]$$ and $$f^{\ast }_{1}=f^{-1}_{1}$$. Given a morphism $$(\,f_{1},f_{2}):(X_{B},{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}} _{B},Y_{B})\longrightarrow (X_{A},{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}} _{A},Y_{A})$$ of $$\tau$$-frames, let $$(X_{B},{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}} _{B},Y_{B})=\hat{F}({\mathcal B})$$, $$(X_{A},{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}} _{A},Y_{A})=\hat{F}({\mathcal A})$$, let also $$(\,f_{1},f_{2})=\hat{F}(\,f)$$, where $$f:{\mathcal A}\longrightarrow{\mathcal B}$$ is the unique homomorphism of the underlying lattices of the $$\tau$$-algebras $${\mathcal A,B}$$ and $$f^{\ast }_{1}:X^{\ast }_{A}\longrightarrow X^{\ast }_{B}$$ be defined as detailed above. We verify below that $$f:{\mathcal A}\longrightarrow{\mathcal B}$$ is a $$\tau$$-algebra homomorphism, i.e. $$f(\oplus _{\delta ,A}(a_{1},\ldots ,a_{n}))=\oplus _{\delta ,B}(\,f(a_{1}),\ldots ,f(a_{n}))$$. By definition of $$\tau$$-frame morphisms (Definition 3.9), $$f^{\ast }_{1}$$ is a $$\tau$$-algebra homomorphism and we then have   \begin{align*} &f(\oplus_{\delta,A}(a_{1},\ldots,a_{n}))=b\\ &\begin{array}{llll} {\hskip-8pt}& \textrm{iff} & f^{\ast}_{1}(\alpha_{A}(\oplus_{\delta,A}(a_{1},\ldots,a_{n})))=\alpha_{B}(b) & \textrm{by definition of f} \\ & \textrm{iff} & f^{\ast}_{1}(\oplus^{\sigma}_{\delta,A}(\alpha_{A}(a_{1}),\ldots,\alpha_{A}(a_{n}))=\alpha_{B}(b) & \textrm{by representation} \\ & \textrm{iff} & \oplus^{\sigma}_{\delta,B}(\,f^{\ast}_{1}(\alpha_{A}(a_{1})),\ldots,f^{\ast}_{1}(\alpha_{A}(a_{n})))=\alpha_{B}(b) & f^{\ast}_{1} \ \textrm{is a homomorphism} \\ & \textrm{iff} & \oplus^{\sigma}_{\delta,B}(\alpha_{B}(\,f(a_{1})),\ldots,\alpha_{B}(\,f(a_{n})))=\alpha_{B}(b)& \textrm{by definition of f} \\ & \textrm{iff} & \alpha_{B}(\oplus_{\delta,B}(\,f(a_{1}),\ldots,f(a_{n})))=\alpha_{B}(b) & \textrm{by representation} \\ & \textrm{iff} & \alpha_{B}(\oplus_{\delta,B}(\,f(a_{1}),\ldots,f(a_{n})))=\alpha_{B}(\,f(\oplus_{\delta,A}(a_{1},\ldots,a_{n}))) & \\ & \textrm{iff} & \oplus_{\delta,B}(\,f(a_{1}),\ldots,f(a_{n}))=f(\oplus_{\delta,A}(a_{1},\ldots,a_{n})) & \textrm{by representation} \end{array} \end{align*}Hence, f is a $$\tau$$-algebra homomorphism and we may define $$\hat{C}(\,f_{1},f_{2})$$ to be the unique $$\tau$$-algebra homomorphism $$f:{\mathcal A}\longrightarrow{\mathcal B}$$, as above, thus extending $$\hat{C}$$ to a contravariant functor from the category of $$\tau$$-frames to the category of $$\tau$$-algebras. Theorem 4.5 (Duality) The contravariant functors $$\hat{F},\hat{C}$$ form a dual equivalence of the categories of normal lattice expansions of similarity type $$\tau$$ with the corresponding categories of $$\tau$$-frames. Proof. For the isomorphism $${\mathcal A}\backsimeq \hat{C}\hat{F}({\mathcal A})=({\mathcal A}_{+})^{+}$$, where $${\mathcal A}_{+}=\hat{F}(A)=(X,{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}} ,Y)$$ is the dual frame of $$\mathcal A$$ and $$X^{\ast }={\mathcal G}_{\kappa o}(X)$$ is the lattice of clopen elements of its canonical extension $${\mathcal G}_{\psi }(X)$$, the results of [19] ensure that $${\mathcal A}, ({\mathcal A}_{+})^{+}$$ are isomorphic as lattices, via the representation map $$\alpha (a)=\{x\in X\;|\; a\in x\}$$. For an n-ary normal lattice operator $$\oplus _{\delta }:{\mathcal A}^{n}\longrightarrow{\mathcal A}$$, where $$\delta \in \tau$$ is of any output type 1, or ∂, it was shown in Proposition 4.1 that $$\alpha (\oplus _{\delta }(a_{1},\ldots ,a_{n}))=\oplus ^{\sigma }_{\delta }(\alpha (a_{1}),\ldots ,\alpha (a_{n}))$$, where the latter is $${\bigcirc}{\kern-6.4pt\mid}\ \,^{\sigma }_{\delta }(\alpha (a_{1}),\ldots ,\alpha (a_{n}))$$, if $$\delta \in \tau _{1}$$ and it is $$\ominus ^{\sigma }_{\delta }(\alpha (a_{1}),\ldots ,\alpha (a_{n}))$$ when $$\delta \in \tau _{\partial }$$, hence the lattice isomorphism $$\alpha$$ is a homomorphism of $$\tau$$-algebras, given also Corollary 3.1. Therefore $${\mathcal A}\backsimeq \hat{C}\hat{F}({\mathcal A})=({\mathcal A}_{+})^{+}$$. For the isomorphism $$\mathfrak{F}\backsimeq \hat{F}\hat{C}(\mathfrak{F})=(\mathfrak{F}^{+})_{+}$$, where $$\mathfrak{F}=(X,{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}} ,Y)$$ and by the results of [19] and by the arguments in the proof of Lemma 4.4, we may as well assume that $$\mathfrak{F}=\hat{F}({\mathcal A})$$, for a unique $$\tau$$-algebra $$\mathcal A$$ such that $${\mathcal A}\backsimeq X^{\ast }$$, where recall that the latter is the $$\tau$$-algebra of clopens of $${\mathcal G}_{\psi }(X)$$. Therefore, $$\hat{F}\hat{C}(\mathfrak{F})=\hat{F}\hat{C}\hat{F}({\mathcal A})\backsimeq \hat{F}({\mathcal A})=\mathfrak{F}$$, given that $$\hat{F},\hat{C}$$ form an adjunction, by the duality of [19]. 5 Modeling full BCK Ono [26–29] studied a number of systems arising from the Gentzen system LJ for intuitionistic logic by dropping a combination of the structural rules of exchange, contraction and weakening (perhaps also association) and expanding the logical signature of the language to include the operator symbols ∘ (fusion, cotenability), $$\leftarrow$$ (reverse implication) and a constant $$\mathfrak{t}$$. The algebraic semantics of these systems has been investigated by Hiroakira Ono, see [29], and others. Following Ono, we let FL be the system with all structural rules dropped, which is precisely the (associative) Full Lambek calculus, and for r ⊆{c, e, w} we designate by $$\textbf{FL}_{r}$$ the system resulting by adding to FL the structural rules in r (where c abbreviates ‘contraction’, e abbreviates ‘exchange’ and similarly for w and ‘weakening’). With the exception of $$\textbf{FL}_{ecw}$$, which is precisely LJ, distribution of conjunctions over disjunctions and conversely does not hold, unless explicitly postulated in the axiomatisation. An FL-algebra is a structure $$\langle L,\leq ,\wedge ,\vee ,0,1,\leftarrow ,\circ ,\rightarrow ,\mathfrak{t}\rangle$$ where (1) ⟨L, ≤, ∧, ∨, 0, 1⟩ is a bounded lattice. (2) $$\langle L,\leq ,\circ ,\mathfrak{t}\rangle$$ is a partially-ordered monoid (∘ is monotone and associative and $$\mathfrak{t}$$ is a two-sided identity element $$a\circ \mathfrak{t}=a=\mathfrak{t}\circ a$$). (3) $$\leftarrow ,\circ ,\rightarrow$$ are residuated, i.e. a ∘ b ≤ c iff $$b\leq a\rightarrow c$$ iff $$a\leq c\leftarrow b$$. (4) for any a ∈ L, a ∘ 0 = 0 = 0 ∘ a. An FL-algebra is known as a residuated lattice. FL-algebras (residuated lattices) are precisely the algebraic models of the (associative) full Lambek calculus. An FL$$_{ew}$$-algebra adds to the axiomatisation the exchange (commutativity) axiom a ∘ b = b ∘ a for the cotenability operator (in which case $$\leftarrow$$ and $$\rightarrow$$ coincide), as well as the weakening axiom b ∘ a ≤ a, in which case combining with commutativity a ∘ b ≤ a ∧ b follows. In addition, by $$1\circ \mathfrak{t}\leq \mathfrak{t}$$, the identity $$\mathfrak{t}=1$$ holds in FL$$_{ew}$$-algebras. FL$$_{ew}$$-algebras are also referred to in the literature as full BCK-algebras, corresponding to full BCK-logic, resulting from BCK whose purely implicational signature is expanded to include conjunction and disjunction connectives, alongside the cotenability logical operator and the constants 0,1. Algebraically, they constitute the class of commutative integral residuated lattices. The language of full BCK is displayed below, where P is a nonempty, countable set of propositional variables.   $$L\ni\varphi\;:=\; p\;(p\in P)\;|\;\top\;|\;\bot\;|\; \varphi\wedge\varphi\;|\;\varphi\vee\varphi\;|\;\varphi\circ\varphi\;|\;\varphi{\rightarrow}\varphi.$$Since we have no interest in this article in studying proof theoretic issues, we may as well assume that the proof system is presented as a symmetric consequence system $$\varphi \vdash \psi$$, directly encoding the corresponding algebraic specification (and thereby being sound and complete in the class of commutative integral residuated lattices). Both operators $$\circ ,\rightarrow$$ are normal, with respective distribution types $$\delta (\circ )=(1,1;1)=\delta$$ and $$\delta (\rightarrow )=(1,\partial ;\partial )={\delta ^{\prime}}$$. This section provides an application of the proposed representation and duality results, modeling the full BCK calculus. BCK-frames are the appropriate $$\tau$$-frames of Section 3, detailed in the sequel, but where we drop, as usual, the topology (axiom F4). There is an option, as pointed out in the Introduction (repeating a point we first made in [20]), as far as interpretations are concerned. The first option considers plain Kripke frames, though two-sorted, and interprets a propositional variable as any stable set V (p) in the full complex algebra $${\mathcal G}_{\psi }(X)$$ of the frame and co-interprets it as a co-stable set in $${\mathcal G}_{\phi }(Y)$$, under the restriction that the co-interpretation is the Galois-dual of the interpretation: $$V{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}(p)=V(p){ }^{{\mathop{=}^{\kern-5pt\shortmid}}}$$. The second option considers general frames and restricts (co)interpretations to assign, respectively, a closed set to a propositional variable. In the general frames approach the received interpretation of the logical operators is typically verbatim the same as in the distributive case, which may be of significant interests in some applied contexts. 5.1 Frames and soundness Definition 5.1 (BCK Frames) A BCK-frame is a $$\tau$$-frame (except for dropping the topology, axiom F4) $$\mathfrak{F}=(X,(R_{\circ },R^{\partial }_{\circ }),{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}} ,Y,(S_>,S^{\partial }_>))$$, for $$\tau =(\delta ,{\delta ^{\prime }})$$, with two additional conditions (E) and (Res). Explicitly, (1) X, Y are nonempty sets and $${\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}$$ ⊆ X × Y is a binary relation generating a Galois connection defined on U ⊆ X and V ⊆ Y by   \begin{align*} \phi(U)=&\,U^{{\mathop{=}^{\kern-5pt\shortmid}}} =\{y\in Y\;|\;\forall u\in U\; u{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}} y\}=\{y\in Y\;|\; U\ {\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\ y\} \\ \psi(V)=&\,{}^{{\mathop{=}^{\kern-5pt\shortmid}}} V =\{x\in X\;|\; \forall v\in V\; x{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}} v\}=\{x\in X\;|\; x\ {\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\ V\} \end{align*}and we let $${\mathcal G}_{\psi }(X),{\mathcal G}_{\phi }(Y)$$ be the complete lattices of stable, $$A=\psi \phi A$$, and co-stable, $$B=\phi \psi B$$, subsets of X and Y, respectively. (F1) The relations x ≤ z iff $$\{x\}^{{\mathop{=}^{\kern-5pt\shortmid}}}\subseteq \{z\}^{{\mathop{=}^{\kern-5pt\shortmid}}}$$ on X and y ≤ v iff $${ }^{{\mathop{=}^{\kern-5pt\shortmid}}}\{y\}\subseteq{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}\{z\}$$ on Y are partial orders on X and of Y, respectively. (F2) $${\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}$$ is increasing in each argument place, i.e. x$${\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}$$y, x ≤ z, y ≤ v imply z$${\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}$$v. (F3) Clopen sets are closed under finite intersections and closed sets are closed under arbitrary intersections, for each of $${\mathcal G}_{\psi }(X),{\mathcal G}_{\phi }(Y)$$ (see Definition 2.3 for the definition of closed, open and clopen elements). (F5) The family of closed sets is the meet-closure of the family of clopen sets, for each of $${\mathcal G}_{\psi }(X),{\mathcal G}_{\phi }(Y)$$. (R1) $$R_{\circ }\;\subseteq X\times X\times X$$ and $$S^{\partial }_>\;\subseteq Y\times X\times Y$$. (R2) $$R_{\circ }, S^{\partial }_>$$ are increasing in the first argument place and decreasing in every other argument place. (R3) For any $$(u_{1},u_{2})\in (X\times X)$$, the set $$R_{\circ } u_{1} u_{2}$$ is a closed element of $${\mathcal G}_{\psi }(X)$$ (in other words, there exists a point z ∈ X such that for any x ∈ X, $$xR_{\circ } u_{1} u_{2}$$ iff z ≤ x) and if both $$u_{i}$$ are clopen, then so is $$R_{\circ } u_{1} u_{2}$$. (R4) For any $$(u_{1},u_{2})\in (X\times Y)$$, the set $$S^{\partial }_> u_{1}u_{2}$$ is a closed element of $${\mathcal G}_{\phi }(Y)$$ (i.e. there is a point v ∈ Y such that for all y ∈ Y, $$yS^{\partial }_> u_{1} u_{2}$$ iff v ≤ y) and if both $$u_{i}$$ are clopen, then so is $$S^{\partial }_> u_{1} u_{2}$$. (R5) $$R^{\partial }_{\circ }\subseteq Y\times (X\times X)$$ and $$\forall (u_{1}, u_{2})\in (X\times X)\;R^{\partial }_{\circ } u_{1} u_{2}\;=\;(R_{\circ } u_{1} u_{2}){ }^{{\mathop{=}^{\kern-5pt\shortmid}}}$$. (R6) $$S_>\subseteq X\times (X\times Y)$$ and $$\forall (v_{1},v_{2})\in (X\times Y)\;S_> v_{1}v_{2}\;=\;{}{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}(S^{\partial }_> v_{1}v_{2})$$. (R7) For all x ∈ X and all $$u_{s},u^{\prime }_{s}\in X$$, for s = 1, 2,   \begin{align*} xR_{\circ} u_{1} (u_{2}\cap u^{\prime}_{2}) &\longrightarrow \forall y\in Y\;(\,yR^{\partial}_{\circ} u_{1} u_{2}\;\wedge\;yR^{\partial}_{\circ} u_{1} u^{\prime}_{2}\;\longrightarrow\;x\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; y) \\ xR_{\circ}(u_{1}\cap u^{\prime}_{1})u_{2} &\longrightarrow \forall y\in Y\;(\,yR^{\partial}_{\circ} u_{1} u_{2}\;\wedge\;yR^{\partial}_{\circ} u^{\prime}_{1} u_{2}\;\longrightarrow\;x\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; y). \end{align*} (R8) For all $$y,u_{2},u^{\prime }_{2}\in Y, u_{1},u^{\prime }_{1}\in X$$  \begin{align*} yS^{\partial}_> u_{1}\left(u_{2}\cap u^{\prime}_{2}\right) &\longrightarrow \forall x\in X\;\left(\,yS_> u_{1} u_{2}\;\wedge\;yS_> u_{1}\cdots u^{\prime}_{2}\;\longrightarrow\;x\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; y\right) \\ yS^{\partial}_> \left(u_{1}\cap u^{\prime}_{1}\right) u_{2}&\longrightarrow\forall x\in X\;\left(\,yS_> u_{1} u_{2}\;\wedge\;yS_{\delta} u^{\prime}_{1} u_{2}\;\longrightarrow\;x\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; y\right). \end{align*} (R9) For all $$(u_{1},u_{2})\in X\times X$$ and z ∈ X, if for all clopen elements $$u^{s}_{*}$$ below $$u_{s}$$, for each s = 1, 2, there exist $$(v_{1},v_{2})\in X\times X$$ such that $$zR_{\delta } v_{1}v_{2}$$ and $$u^{s}_{*}\leq v_{s}$$, for each s = 1, 2, then there exist $$(v^{\prime }_{1},v^{\prime }_{2})\in X\times X$$ such that $$zR_{\delta } v^{\prime }_{1} v^{\prime }_{2}$$ and $$u_{s}\leq v^{\prime }_{s}$$, for each s = 1, 2. (R10) For all $$(u_{1},u_{2})\in X\times Y$$ and y ∈ Y, if for all clopen elements $$u^{s}_{*}$$ below $$u_{s}$$, for each s = 1, 2, there exist $$(v_{1},v_{2})\in X\times Y$$ such that $$yS^{\partial }_{\delta ^{\prime }} v_{1} v_{2}$$ and $$u^{s}_{*}\leq v_{s}$$, for each s = 1, 2, then there exist $$(v^{\prime }_{1},v^{\prime }_{2})\in X\times Y$$ such that $$yS^{\partial }_{\delta ^{\prime }} v^{\prime }_{1} v^{\prime }_{2}$$ and $$u_{s}\leq v^{\prime }_{s}$$, for each s = 1, 2. (E) For all u, x, z ∈ X, $$\;uR_{\circ } xz\ \textrm{iff}\; uR_{\circ } zx$$. (Res) The following are equivalent, where $$A,B,C\in{\mathcal G}_{\psi }(X)$$ are stable sets $$\forall u\in X\;\forall x\in A\;\forall z\in B\;(uR_{\circ } xz\;\Longrightarrow \;u\in C)$$ $$\forall z\in B\;\forall y\in Y\;\forall v\in X\;\forall w\in Y\;(yS^{\partial }_> vw\;\wedge \;v\in A\;\wedge \;C{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}} w\;\Longrightarrow \;z{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}} y)$$. Note that we have not listed axiom (R8) for $$\tau$$-frames as it refers to distribution types the kind of which does not occur in the similarity type $$\tau$$ under consideration. Definition 5.2 For $$U,U^{\prime }\subseteq X$$, $$A,A^{\prime }\in{\mathcal G}_{\psi }(X)$$ and $$V,V^{\prime }\subseteq Y$$, $$B,B^{\prime }\in{\mathcal G}_{\phi }(Y)$$ define the operators   $${\hskip-10pt}\widehat{\bigcirc}{\kern-6.4pt\mid}\ \,_{\circ}(U,U^{\prime}) = \{x\in X\;|\;\exists u,u^{\prime}\in X\;(xR_{\circ} uu^{\prime}\;\wedge\;u\in U\;\wedge\;u^{\prime}\in U^{\prime})\}$$ (15)  $${\hskip-160pt}{\bigcirc}{\kern-6.4pt\mid}\ \,_{\circ}(A,A^{\prime}) = \psi\phi\widehat{\bigcirc}{\kern-6.4pt\mid}\ \,_{\circ}(A,A^{\prime})$$ (16)  $${\hskip-150pt}{\bigcirc}{\kern-6.4pt\mid}\ \,^{\partial}_{\circ}(B,B^{\prime}) = \phi{\bigcirc}{\kern-6.4pt\mid}\ \,_{\circ}(\psi B,\psi B^{\prime})$$ (17)  \begin{align} \widehat{\ominus}^{\partial}_>(V,V^{\prime}) = \{y\in Y\;|\; \exists u\in X\;\exists v\in Y\;(yS^{\partial}_>uv\;\wedge\;u\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; V\;\wedge\; v\in V^{\prime})\} \end{align} (18)  $${\hskip-160pt}\ominus^{\partial}_>(B,B^{\prime}) = \phi\psi\widehat{\ominus}^{\partial}_>(B,B^{\prime})$$ (19)  $${\hskip-145pt}\ominus_>(A,A^{\prime}) = \psi\ominus^{\partial}_>(\phi A,\phi A^{\prime}).$$ (20)Simplify notation by writing $$A\,{\bigcirc}{\kern-6.4pt\mid}\ \,\, A^{\prime }$$, $$B\,{\bigcirc}{\kern-6.4pt\mid}\ \,^{\partial } B^{\prime }$$ for $${\bigcirc}{\kern-6.4pt\mid}\ \,_{\circ }(A,A^{\prime }), {\bigcirc}{\kern-6.4pt\mid}\ \,^{\partial }_{\circ }(B,B^{\prime })$$, respectively. Similarly, write $$A\Rightarrow A^{\prime }, B\Rightarrow ^{\partial } B^{\prime }$$ for $$\ominus _>(A,A^{\prime }), \ominus ^{\partial }_>(B,B^{\prime })$$ respectively, recalling that $$B \ {\bigcirc}{\kern-6.4pt\mid}\ \,^{\partial } B^{\prime }=\phi (\psi B\,{\bigcirc}{\kern-6.4pt\mid}\ \,\, \psi B^{\prime })$$ while also $$A\Rightarrow A^{\prime }=\psi (\phi A\Rightarrow ^{\partial }\phi A^{\prime })$$. In the following lemma we prove properties of the operators that will be used in the sequel. Lemma 5.3 By the results of Section 3 the following hold. (1) For any x, z ∈ X, y ∈ Y,   \begin{align*} \widehat{\bigcirc}{\kern-6.4pt\mid}\ \,_{\circ}(\Gamma x,\Gamma z)=&\,(\Gamma x){\bigcirc}{\kern-6.4pt\mid}\ \,(\Gamma z) =R_{\circ} xz\in{\mathcal G}_{\kappa}(X)\\ \widehat{\ominus}^{\partial}_>\left(\{x\}{{}^{{\mathop{=}^{\kern-5pt\shortmid}}}},\Gamma y\right)=&\,\left(\{x\}{{}^{{\mathop{=}^{\kern-5pt\shortmid}}}}\right)\Rightarrow^{\partial}(\Gamma y)=S^{\partial}_>xy\in{\mathcal G}_{\kappa}(Y). \end{align*} (2) The operators $$\widehat{\bigcirc}{\kern-6.4pt\mid}\ \,_{\circ }, \widehat{\ominus }_>$$ restrict to operators on clopen elements of $${\mathcal G}_{\psi }(X)$$ and, thereby, so do the operators $${\bigcirc}{\kern-6.4pt\mid}\ \,,\Rightarrow$$. (3) $$B\ {\bigcirc}{\kern-6.4pt\mid}\ \,^{\partial } B^{\prime }=\{y\in Y\;|\;\forall u,u^{\prime }\in X\;(u{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}} B\;\wedge \;u^{\prime }{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; B^{\prime }\;\Longrightarrow \;yR^{\partial }_{\circ } uu^{\prime })\}$$. (4) $$A\Rightarrow A^{\prime } =\{x\in X\;|\;\forall u\in X\;\forall v\in Y\;(u\in A\;\wedge \; A^{\prime }{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}} v\;\Longrightarrow \;xS_>uv)\}$$. (5) $$A{\bigcirc}{\kern-6.4pt\mid}\ \,A^{\prime }=\bigvee ^{x\in A}_{z\in A^{\prime }}(\Gamma x\ {\bigcirc}{\kern-6.4pt\mid}\ \,\ \Gamma z)$$ and, similarly, $$B\Rightarrow ^{\partial } B^{\prime }=\bigvee ^{x{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}} B}_{v\in B^{\prime }}(\{x\}{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}\Rightarrow ^{\partial }\Gamma v)$$. (6) The operators $${\bigcirc}{\kern-6.4pt\mid}\ \,,\Rightarrow$$ are the $$\sigma$$-extensions of their restrictions to clopens and hence they are completely normal operators in the full complex algebra of the frame, of distribution types $$\delta =(1,1;1)$$ and $${\delta ^{\prime }}=(1,\partial ;\partial )$$, respectively. Proof. 1) is an instance of Lemma 3.3 and 2) is one of Corollary 3.4, while 3) and 4) follow by Lemma 3.6 and 5) follows by Corollary 3.7. Recall that only the F axioms and axioms R1–R4 are used in the proofs of Lemmas 3.3, 3.6 and Corollaries 3.4, 3.7 are consequences of Lemma 3.3. Recall also that the definitions of the dual relations were introduced in the proof of Lemma 3.6 and were subsequently included as frame axioms (R5, R6) in Section 3.2. 6) follows by Proposition 3.13, whose proof relied on the frame axioms (R7–R10). Lemma 5.4 The frame conditions (E,Res) ensure commutativity $$A\,{\bigcirc}{\kern-6.4pt\mid}\ \,\, A^{\prime }=A^{\prime }\,{\bigcirc}{\kern-6.4pt\mid}\ \,\, A$$ and residuation $$A\,{\bigcirc}{\kern-6.4pt\mid}\ \,\, B\subseteq C$$ iff B ⊆ A ⇒ C. Proof. By Lemma 5.3, $$A\ {\bigcirc}{\kern-6.4pt\mid}\ \,\ A^{\prime }=\bigvee ^{x\in A}_{z\in A^{\prime }}(\Gamma x\ {\bigcirc}{\kern-6.4pt\mid}\ \,\ \Gamma z)=\bigvee ^{x\in A}_{z\in A^{\prime }}R_{\circ } xz =\bigvee ^{x\in A}_{z\in A^{\prime }}R_{\circ } zx=\bigvee ^{x\in A}_{z\in A^{\prime }}(\Gamma z\ {\bigcirc}{\kern-6.4pt\mid}\ \,\ \Gamma x)=A^{\prime }\ {\bigcirc}{\kern-6.4pt\mid}\ \,\ A$$. For residuation, $$A\,{\bigcirc}{\kern-6.4pt\mid}\ \,\, B\subseteq C$$ iff $$\,\bigvee ^{x\in A}_{z\in B}R_{\circ } xz\subseteq C$$ iff $$\forall x\in A\;\forall z\in B\; R_{\circ } xz\subseteq C$$. Note also that   \begin{align*} A\Rightarrow C&= \psi(\phi A\Rightarrow^{\partial}\phi C)\\ &=\psi\phi\psi\{\,y\in Y\;|\;\exists v\in X\exists w\in Y\;(yS^{\partial}_>vw\;\wedge\;v\in A\;\wedge\;C\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; w)\}\\ &=\psi\{\,y\in Y\;|\;\exists v\in X\exists w\in Y\;(yS^{\partial}_>vw\;\wedge\;v\in A\;\wedge\;C\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; w)\} \end{align*}hence   \begin{align*} B\subseteq A\Rightarrow C &\textrm{iff}\ \ B\subseteq \psi\{\,y\in Y\;|\;\exists v\in X\exists w\in Y\;(yS^{\partial}_>vw\;\wedge\;v\in A\;\wedge\;C\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; w)\}\\ &\textrm{iff}\ \ B \ {\mathop{=}^{\kern-5pt\raise-4pt\shortmid}} \ \{\,y\in Y\;|\;\exists v\in X\exists w\in Y\;(yS^{\partial}_>vw\;\wedge\;v\in A\;\wedge\;C\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; w)\}\\ &\textrm{iff}\ \ \forall z\in B\;\forall y\in Y\;(\exists v\in X\exists w\in Y\;(\,yS^{\partial}_>vw\;\wedge\;v\in A\;\wedge\;C\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; w)\;\Longrightarrow\;z\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; y). \end{align*}Therefore, the frame condition (Res) is equivalent to the residuation condition for $${\bigcirc}{\kern-6.4pt\mid}\ \,,\Rightarrow$$. 5.2 Models, soundness and completeness BCK models $$\mathfrak{M}=(\mathfrak{F},V,V{ }^{{\mathop{=}^{\kern-5pt\shortmid}}})$$ consist of a BCK-frame $$\mathfrak{F}=(X,(R_{\circ },R^{\partial }_{\circ }),{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}} ,Y,(S_>,S^{\partial }_>))$$ together with an interpretation V and a co-interpretation $$V{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}$$ assigning to a propositional variable a stable and a co-stable set, respectively, satisfying $$V{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}(p)=V(p){ }^{{\mathop{=}^{\kern-5pt\shortmid}}}$$. As pointed out in [20], there appear to exist two options in defining models and in each case soundness and completeness can be established. According to the first option, V assigns any stable set $$V(p)\in{\mathcal G}_{\psi }(X)$$ to a propositional variable p, but the resulting interpretation clauses are non-standard (see for example [8] where the fusion-implication fragment of FL is modeled). Alternatively, we may restrict interpretations to assign a closed set $$V(p)\in{\mathcal G}_{\kappa }(X)\subseteq{\mathcal G}_{\psi }(X)$$, in which case the interpretation pattern follows lines familiar from the distributive case. The difference then is one of working with plain (two-sorted) Kripke frames, in which case the intended interpretation of some operators may be lost, or working with general frames (restricting interpretations to assign closed elements to propositional variables), in which case the intended meaning of operators is re-captured. The relational representation of operators and subsequent Stone duality we have presented can be used in each of the above cases and, for explicitness, we present both approaches in modeling BCK. 5.2.1 Plain (two-sorted) Kripke frames and models Let $$\mathfrak{M}=(\mathfrak{F},V,V{ }^{{\mathop{=}^{\kern-5pt\shortmid}}})$$ be a frame $$\mathfrak{F}=(X,R_{\circ },S_>,{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}} ,R^{\partial }_{\circ },S^{\partial }_>,Y)$$ together with an interpretation V assigning a stable set $$V(p)\in{\mathcal G}_{\psi }(X)$$ and a co-interpretation $$V{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}$$ assigning a costable set $$V{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}(p)\in{\mathcal G}_{\phi }(Y)$$ such that $$V{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}(p)=V(p){ }^{{\mathop{=}^{\kern-5pt\shortmid}}}$$. Extend the interpretation and co-interpretation recursively to all sentences as in Table 1, where R is the complement of the Galois relation $${\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}$$ of the frame. Table 1 Interpretation and co-interpretation of full BCK in plain Kripke frames $$[\![p]\!]$$  = V (p)    $$[\![\top]\!]$$  = X    $$[\![\bot]\!]$$  $$=\varnothing _{\psi }$$  $$=\bigcap{\mathcal G}_{\psi }(X)$$  $$[\![\vartheta \wedge \chi ]\!]$$  $$=[\![\vartheta ]\!]\cap [\![\chi ]\!]$$    $$[\![\vartheta \vee \chi ]\!]$$  $$=\psi \phi ([\![\vartheta ]\!]\cup [\![\chi ]\!])$$  $$=\{x\in X\;|\; \forall y\in Y\;(xRy\;\longrightarrow y\not \in (\!|\vartheta |\!)\;\vee \;y\not \in (\!|\chi |\!))\}$$  $$[\![\vartheta \circ \chi ]\!]$$  $$=[\![\vartheta ]\!]{\bigcirc}{\kern-6.4pt\mid}\ \,[\![\chi ]\!]$$  $$=\psi \phi \{x\in X\;|\;\exists z,z^{\prime }\in X\;(xR_{\circ } zz^{\prime }\;\wedge \;z\in [\![\vartheta ]\!]\;\wedge \;z^{\prime }\in [\![\chi ]\!])\}$$  $$[\![\vartheta \rightarrow \chi ]\!]$$  $$=[\![\vartheta ]\!]\Rightarrow [\![\chi ]\!]$$  $$=\{x\in X\;|\; \forall z\in X\;\forall y\in Y;(z\in [\![\vartheta ]\!]\;\wedge \;y\in (\!|\chi |\!) \;\longrightarrow \;xS_>zy)\}$$  $$(\!|p|\!)$$  $$=[\![p]\!]{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}$$    $$(\!|\top|\!)$$  $$=\varnothing _{\phi }$$  $$=\;\bigcap{\mathcal G}_{\phi }(Y)$$  $$(\!|\bot|\!)$$  = Y    $$(\!|\vartheta \wedge \chi |\!)$$  $$=\phi \psi ((\!|\vartheta |\!) \cup (\!|\chi |\!) )$$  $$=\;\{y\in Y\;|\;\forall x\in X\;(xRy\;\longrightarrow \;x\not \in [\![\vartheta ]\!]\;\vee \;x\not \in [\![\chi ]\!])\}$$  $$(\!|\vartheta \vee \chi |\!)$$  $$=(\!|\vartheta |\!)\cap (\!|\chi |\!)$$    $$(\!|\vartheta \circ \chi |\!)$$  $$=(\!|\vartheta |\!) {\bigcirc}{\kern-6.4pt\mid}\ \,^{\partial }(\!|\chi |\!)$$  $$=\{y\in Y\;|\;\forall x,z\in X\;(x\in [\![\vartheta ]\!]\;\wedge \;z\in [\![\chi ]\!]\;\longrightarrow \; yR^{\partial }_{\circ } xz)\}$$  $$(\!|\vartheta \rightarrow \chi |\!)$$  $$=(\!|\vartheta |\!) \Rightarrow ^{\partial }(\!|\chi |\!)$$  $$=\;\phi \psi \{y\in Y\;|\;\exists x\in X\exists v\in Y\; (yS^{\partial }_> xv\;\wedge \;x\in [\![\vartheta ]\!]\;\wedge \;v\in (\!|\chi |\!) )\}$$  $$[\![p]\!]$$  = V (p)    $$[\![\top]\!]$$  = X    $$[\![\bot]\!]$$  $$=\varnothing _{\psi }$$  $$=\bigcap{\mathcal G}_{\psi }(X)$$  $$[\![\vartheta \wedge \chi ]\!]$$  $$=[\![\vartheta ]\!]\cap [\![\chi ]\!]$$    $$[\![\vartheta \vee \chi ]\!]$$  $$=\psi \phi ([\![\vartheta ]\!]\cup [\![\chi ]\!])$$  $$=\{x\in X\;|\; \forall y\in Y\;(xRy\;\longrightarrow y\not \in (\!|\vartheta |\!)\;\vee \;y\not \in (\!|\chi |\!))\}$$  $$[\![\vartheta \circ \chi ]\!]$$  $$=[\![\vartheta ]\!]{\bigcirc}{\kern-6.4pt\mid}\ \,[\![\chi ]\!]$$  $$=\psi \phi \{x\in X\;|\;\exists z,z^{\prime }\in X\;(xR_{\circ } zz^{\prime }\;\wedge \;z\in [\![\vartheta ]\!]\;\wedge \;z^{\prime }\in [\![\chi ]\!])\}$$  $$[\![\vartheta \rightarrow \chi ]\!]$$  $$=[\![\vartheta ]\!]\Rightarrow [\![\chi ]\!]$$  $$=\{x\in X\;|\; \forall z\in X\;\forall y\in Y;(z\in [\![\vartheta ]\!]\;\wedge \;y\in (\!|\chi |\!) \;\longrightarrow \;xS_>zy)\}$$  $$(\!|p|\!)$$  $$=[\![p]\!]{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}$$    $$(\!|\top|\!)$$  $$=\varnothing _{\phi }$$  $$=\;\bigcap{\mathcal G}_{\phi }(Y)$$  $$(\!|\bot|\!)$$  = Y    $$(\!|\vartheta \wedge \chi |\!)$$  $$=\phi \psi ((\!|\vartheta |\!) \cup (\!|\chi |\!) )$$  $$=\;\{y\in Y\;|\;\forall x\in X\;(xRy\;\longrightarrow \;x\not \in [\![\vartheta ]\!]\;\vee \;x\not \in [\![\chi ]\!])\}$$  $$(\!|\vartheta \vee \chi |\!)$$  $$=(\!|\vartheta |\!)\cap (\!|\chi |\!)$$    $$(\!|\vartheta \circ \chi |\!)$$  $$=(\!|\vartheta |\!) {\bigcirc}{\kern-6.4pt\mid}\ \,^{\partial }(\!|\chi |\!)$$  $$=\{y\in Y\;|\;\forall x,z\in X\;(x\in [\![\vartheta ]\!]\;\wedge \;z\in [\![\chi ]\!]\;\longrightarrow \; yR^{\partial }_{\circ } xz)\}$$  $$(\!|\vartheta \rightarrow \chi |\!)$$  $$=(\!|\vartheta |\!) \Rightarrow ^{\partial }(\!|\chi |\!)$$  $$=\;\phi \psi \{y\in Y\;|\;\exists x\in X\exists v\in Y\; (yS^{\partial }_> xv\;\wedge \;x\in [\![\vartheta ]\!]\;\wedge \;v\in (\!|\chi |\!) )\}$$  Table 1 Interpretation and co-interpretation of full BCK in plain Kripke frames $$[\![p]\!]$$  = V (p)    $$[\![\top]\!]$$  = X    $$[\![\bot]\!]$$  $$=\varnothing _{\psi }$$  $$=\bigcap{\mathcal G}_{\psi }(X)$$  $$[\![\vartheta \wedge \chi ]\!]$$  $$=[\![\vartheta ]\!]\cap [\![\chi ]\!]$$    $$[\![\vartheta \vee \chi ]\!]$$  $$=\psi \phi ([\![\vartheta ]\!]\cup [\![\chi ]\!])$$  $$=\{x\in X\;|\; \forall y\in Y\;(xRy\;\longrightarrow y\not \in (\!|\vartheta |\!)\;\vee \;y\not \in (\!|\chi |\!))\}$$  $$[\![\vartheta \circ \chi ]\!]$$  $$=[\![\vartheta ]\!]{\bigcirc}{\kern-6.4pt\mid}\ \,[\![\chi ]\!]$$  $$=\psi \phi \{x\in X\;|\;\exists z,z^{\prime }\in X\;(xR_{\circ } zz^{\prime }\;\wedge \;z\in [\![\vartheta ]\!]\;\wedge \;z^{\prime }\in [\![\chi ]\!])\}$$  $$[\![\vartheta \rightarrow \chi ]\!]$$  $$=[\![\vartheta ]\!]\Rightarrow [\![\chi ]\!]$$  $$=\{x\in X\;|\; \forall z\in X\;\forall y\in Y;(z\in [\![\vartheta ]\!]\;\wedge \;y\in (\!|\chi |\!) \;\longrightarrow \;xS_>zy)\}$$  $$(\!|p|\!)$$  $$=[\![p]\!]{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}$$    $$(\!|\top|\!)$$  $$=\varnothing _{\phi }$$  $$=\;\bigcap{\mathcal G}_{\phi }(Y)$$  $$(\!|\bot|\!)$$  = Y    $$(\!|\vartheta \wedge \chi |\!)$$  $$=\phi \psi ((\!|\vartheta |\!) \cup (\!|\chi |\!) )$$  $$=\;\{y\in Y\;|\;\forall x\in X\;(xRy\;\longrightarrow \;x\not \in [\![\vartheta ]\!]\;\vee \;x\not \in [\![\chi ]\!])\}$$  $$(\!|\vartheta \vee \chi |\!)$$  $$=(\!|\vartheta |\!)\cap (\!|\chi |\!)$$    $$(\!|\vartheta \circ \chi |\!)$$  $$=(\!|\vartheta |\!) {\bigcirc}{\kern-6.4pt\mid}\ \,^{\partial }(\!|\chi |\!)$$  $$=\{y\in Y\;|\;\forall x,z\in X\;(x\in [\![\vartheta ]\!]\;\wedge \;z\in [\![\chi ]\!]\;\longrightarrow \; yR^{\partial }_{\circ } xz)\}$$  $$(\!|\vartheta \rightarrow \chi |\!)$$  $$=(\!|\vartheta |\!) \Rightarrow ^{\partial }(\!|\chi |\!)$$  $$=\;\phi \psi \{y\in Y\;|\;\exists x\in X\exists v\in Y\; (yS^{\partial }_> xv\;\wedge \;x\in [\![\vartheta ]\!]\;\wedge \;v\in (\!|\chi |\!) )\}$$  $$[\![p]\!]$$  = V (p)    $$[\![\top]\!]$$  = X    $$[\![\bot]\!]$$  $$=\varnothing _{\psi }$$  $$=\bigcap{\mathcal G}_{\psi }(X)$$  $$[\![\vartheta \wedge \chi ]\!]$$  $$=[\![\vartheta ]\!]\cap [\![\chi ]\!]$$    $$[\![\vartheta \vee \chi ]\!]$$  $$=\psi \phi ([\![\vartheta ]\!]\cup [\![\chi ]\!])$$  $$=\{x\in X\;|\; \forall y\in Y\;(xRy\;\longrightarrow y\not \in (\!|\vartheta |\!)\;\vee \;y\not \in (\!|\chi |\!))\}$$  $$[\![\vartheta \circ \chi ]\!]$$  $$=[\![\vartheta ]\!]{\bigcirc}{\kern-6.4pt\mid}\ \,[\![\chi ]\!]$$  $$=\psi \phi \{x\in X\;|\;\exists z,z^{\prime }\in X\;(xR_{\circ } zz^{\prime }\;\wedge \;z\in [\![\vartheta ]\!]\;\wedge \;z^{\prime }\in [\![\chi ]\!])\}$$  $$[\![\vartheta \rightarrow \chi ]\!]$$  $$=[\![\vartheta ]\!]\Rightarrow [\![\chi ]\!]$$  $$=\{x\in X\;|\; \forall z\in X\;\forall y\in Y;(z\in [\![\vartheta ]\!]\;\wedge \;y\in (\!|\chi |\!) \;\longrightarrow \;xS_>zy)\}$$  $$(\!|p|\!)$$  $$=[\![p]\!]{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}$$    $$(\!|\top|\!)$$  $$=\varnothing _{\phi }$$  $$=\;\bigcap{\mathcal G}_{\phi }(Y)$$  $$(\!|\bot|\!)$$  = Y    $$(\!|\vartheta \wedge \chi |\!)$$  $$=\phi \psi ((\!|\vartheta |\!) \cup (\!|\chi |\!) )$$  $$=\;\{y\in Y\;|\;\forall x\in X\;(xRy\;\longrightarrow \;x\not \in [\![\vartheta ]\!]\;\vee \;x\not \in [\![\chi ]\!])\}$$  $$(\!|\vartheta \vee \chi |\!)$$  $$=(\!|\vartheta |\!)\cap (\!|\chi |\!)$$    $$(\!|\vartheta \circ \chi |\!)$$  $$=(\!|\vartheta |\!) {\bigcirc}{\kern-6.4pt\mid}\ \,^{\partial }(\!|\chi |\!)$$  $$=\{y\in Y\;|\;\forall x,z\in X\;(x\in [\![\vartheta ]\!]\;\wedge \;z\in [\![\chi ]\!]\;\longrightarrow \; yR^{\partial }_{\circ } xz)\}$$  $$(\!|\vartheta \rightarrow \chi |\!)$$  $$=(\!|\vartheta |\!) \Rightarrow ^{\partial }(\!|\chi |\!)$$  $$=\;\phi \psi \{y\in Y\;|\;\exists x\in X\exists v\in Y\; (yS^{\partial }_> xv\;\wedge \;x\in [\![\vartheta ]\!]\;\wedge \;v\in (\!|\chi |\!) )\}$$  Let $$\Vdash , \Vdash ^{\!\!\partial }$$ be defined by $$x\Vdash \vartheta$$ iff $$x\in [\![\vartheta ]\!]\in{\mathcal G}_{\psi }(X)$$ and $$y\Vdash ^{\!\!\partial }\vartheta$$ iff $$y\in (\!|\vartheta |\!)\in{\mathcal G}_{\phi }(Y)$$. Unfolding definitions and recalling that R is the complement of the Galois relation $${\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}$$ of the frame we obtain in particular for fusion and implication the following (co)satisfaction clauses, where $$\overline{S}_>$$ is the complement of $$S_>$$ and we let $$\tilde{S}^{\partial }_>=RS^{\partial }_>$$ designate the composition of R with $$S^{\partial }_>$$ and $$\tilde{R}_{\circ }=R^{-1}R_{\circ }$$ stand for the composition of the converse $$R^{-1}$$ of R with $$R_{\circ }$$.   \begin{array}{lll} X\ni u\Vdash\vartheta\circ\chi & \textrm{iff} & \quad \forall y\in Y\;\left(uRy\;\longrightarrow\;\exists z,z^{\prime}\in X\;\left(\,y\tilde{R}_{\circ} zz^{\prime}\;\wedge z\Vdash\vartheta\;\wedge\;z^{\prime}\Vdash\chi\right)\right) \\ Y\ni y\Vdash^{\!\!\partial}\vartheta\circ\chi & \textrm{iff} & \quad \forall x,z\in X\;\left(x\Vdash\vartheta\;\wedge\;z\Vdash\chi\;\longrightarrow\;yR^{\partial}_{\circ} xz\right) \\ X\ni u\Vdash\vartheta\! \rightarrow \!\chi & \textrm{iff} & \quad \forall x\in X\;\forall y\in Y\;(x\Vdash\vartheta\;\wedge\; u\overline{S}_>xy\;\longrightarrow\;y\not\Vdash^{\!\!\partial}\chi) \\ Y\in y\Vdash^{\!\!\partial}\vartheta\! \rightarrow \!\chi & \textrm{iff} & \quad \forall z\in X\;\left(zRy\;\longrightarrow\;\exists x\in X\;\exists v\in Y\;\left(x\Vdash\vartheta\;\wedge\;z\tilde{ S}^{\partial}_>xv\;\wedge\;v\Vdash^{\!\!\partial}\chi\right)\right). \end{array} The interested reader may wish to compare the above clauses with the corresponding clauses in the generalized Kripke frames approach of [8], where fusion and implication are modeled. Satisfiability and validity of a sentence at a state, frame or class of frames is defined as usual. A sequent $$\varphi \vdash \psi$$ is valid, written $$\vartheta \Vdash \chi$$, iff for any model and state x ∈ X in the underlying frame of the model, if $$x\Vdash \vartheta$$, then $$x\Vdash \chi$$, or, equivalently, if $$y\Vdash ^{\!\!\partial }\chi$$, then $$y\Vdash ^{\!\!\partial }\vartheta$$, for any y ∈ Y. Theorem 5.5 (Soundness and completeness) Full BCK is sound and complete in plain, two-sorted Kripke frame semantics. Proof. We have established the distribution properties of $$\, {\bigcirc}{\kern-6.4pt\mid}\ \,$$ and ⇒ in Lemma 5.3 and commutativity of $${\bigcirc}{\kern-6.4pt\mid}\ \,$$ and residuation of $${\bigcirc}{\kern-6.4pt\mid}\ \,$$ with ⇒ have been established in Lemma 5.4. This proves soundness of BCK. Completeness follows from the representation theorem proven in Section 4. 5.2.2 General (two-sorted) frames and models General (two-sorted) frames are like plain two-sorted frames except that they restrict interpretations and co-interpretations to assign closed elements of $${\mathcal G}_{\psi }(X)$$ and $${\mathcal G}_{\phi }(Y)$$, respectively, to propositional variables. The recursive extension to an interpretation [[ ]] and a co-interpretation (| |) is given by the same clauses as in plain frames, except that taking the Galois closure $$\psi \phi$$ in the cases of the interpretation of fusion and $$\phi \psi$$ for the co-interpretation of implication become redundant as the respective sets are already Galois closed (stable). The satisfaction and co-satisfaction (refutation) relations can be specified by the clauses in Table 2. Table 2 Interpretation and co-interpretation of full BCK in general Kripke frames (x, u, z ∈ X, y, v ∈ Y, R = $${\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}$$?, $$\varnothing _{\psi }=\bigcap{\mathcal G}_{\psi }(X),\;\varnothing _{\phi }=\bigcap{\mathcal G}_{\phi }(Y)$$)  $$x\Vdash p$$  iff  x ∈ V (p)  $$y\Vdash ^{\!\!\partial } p$$  iff  $$y\in V{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}(p)$$  $$x\Vdash \top$$    always  $$y\Vdash ^{\!\!\partial }\bot$$    always  $$x\Vdash \bot$$  iff  $$x\in \varnothing _{\psi }$$  $$y\Vdash ^{\!\!\partial }\top$$  iff  $$y\in \varnothing _{\phi }$$  $$x\Vdash \varphi \wedge \psi$$  iff  $$x\Vdash \varphi$$ and $$x\Vdash \psi$$  $$y\Vdash ^{\!\!\partial }\varphi \vee \psi$$  iff  $$y\Vdash ^{\!\!\partial }\varphi$$ and $$y\Vdash ^{\!\!\partial }\psi$$  $$x\Vdash \varphi \vee \psi$$  iff  $$\forall y\;(xRy\;\Longrightarrow \;y\not \Vdash ^{\!\!\partial }\varphi \ \textrm{or}\ y\not \Vdash ^{\!\!\partial }\psi )$$  $$y\Vdash ^{\!\!\partial }\varphi \wedge \psi$$  iff  $$\forall x\;(xRy\;\Longrightarrow \;x\not \Vdash \varphi \ \textrm{or}\ x\not \Vdash \psi )$$  $$x\Vdash \varphi \circ \psi$$  iff  $$\exists u, z\;(xR_{\circ } uz\ \textrm{and}\ u\Vdash \varphi \ \textrm{and}\ z\Vdash \psi )$$  $$y\Vdash ^{\!\!\partial }\varphi \circ \psi$$  iff  $$\forall x,z\;(x\Vdash \varphi \ \textrm{and}\ z\Vdash \psi \;\Longrightarrow \; yR^{\partial }_{\circ } xz)$$  $$x\Vdash \varphi \rightarrow \psi$$  iff  $$\forall z\forall y\;(z\Vdash \varphi \ \textrm{and}\ y\Vdash ^{\!\!\partial }\psi \;\Longrightarrow \;xS_>zy)$$  $$y\Vdash ^{\!\!\partial }\varphi \rightarrow \psi$$  iff  $$\exists x, v\;(yR^{\partial }_>xv\ \textrm{and}\ x\Vdash \varphi \ \textrm{but}\ v\Vdash ^{\!\!\partial }\psi )$$  (x, u, z ∈ X, y, v ∈ Y, R = $${\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}$$?, $$\varnothing _{\psi }=\bigcap{\mathcal G}_{\psi }(X),\;\varnothing _{\phi }=\bigcap{\mathcal G}_{\phi }(Y)$$)  $$x\Vdash p$$  iff  x ∈ V (p)  $$y\Vdash ^{\!\!\partial } p$$  iff  $$y\in V{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}(p)$$  $$x\Vdash \top$$    always  $$y\Vdash ^{\!\!\partial }\bot$$    always  $$x\Vdash \bot$$  iff  $$x\in \varnothing _{\psi }$$  $$y\Vdash ^{\!\!\partial }\top$$  iff  $$y\in \varnothing _{\phi }$$  $$x\Vdash \varphi \wedge \psi$$  iff  $$x\Vdash \varphi$$ and $$x\Vdash \psi$$  $$y\Vdash ^{\!\!\partial }\varphi \vee \psi$$  iff  $$y\Vdash ^{\!\!\partial }\varphi$$ and $$y\Vdash ^{\!\!\partial }\psi$$  $$x\Vdash \varphi \vee \psi$$  iff  $$\forall y\;(xRy\;\Longrightarrow \;y\not \Vdash ^{\!\!\partial }\varphi \ \textrm{or}\ y\not \Vdash ^{\!\!\partial }\psi )$$  $$y\Vdash ^{\!\!\partial }\varphi \wedge \psi$$  iff  $$\forall x\;(xRy\;\Longrightarrow \;x\not \Vdash \varphi \ \textrm{or}\ x\not \Vdash \psi )$$  $$x\Vdash \varphi \circ \psi$$  iff  $$\exists u, z\;(xR_{\circ } uz\ \textrm{and}\ u\Vdash \varphi \ \textrm{and}\ z\Vdash \psi )$$  $$y\Vdash ^{\!\!\partial }\varphi \circ \psi$$  iff  $$\forall x,z\;(x\Vdash \varphi \ \textrm{and}\ z\Vdash \psi \;\Longrightarrow \; yR^{\partial }_{\circ } xz)$$  $$x\Vdash \varphi \rightarrow \psi$$  iff  $$\forall z\forall y\;(z\Vdash \varphi \ \textrm{and}\ y\Vdash ^{\!\!\partial }\psi \;\Longrightarrow \;xS_>zy)$$  $$y\Vdash ^{\!\!\partial }\varphi \rightarrow \psi$$  iff  $$\exists x, v\;(yR^{\partial }_>xv\ \textrm{and}\ x\Vdash \varphi \ \textrm{but}\ v\Vdash ^{\!\!\partial }\psi )$$  Table 2 Interpretation and co-interpretation of full BCK in general Kripke frames (x, u, z ∈ X, y, v ∈ Y, R = $${\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}$$?, $$\varnothing _{\psi }=\bigcap{\mathcal G}_{\psi }(X),\;\varnothing _{\phi }=\bigcap{\mathcal G}_{\phi }(Y)$$)  $$x\Vdash p$$  iff  x ∈ V (p)  $$y\Vdash ^{\!\!\partial } p$$  iff  $$y\in V{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}(p)$$  $$x\Vdash \top$$    always  $$y\Vdash ^{\!\!\partial }\bot$$    always  $$x\Vdash \bot$$  iff  $$x\in \varnothing _{\psi }$$  $$y\Vdash ^{\!\!\partial }\top$$  iff  $$y\in \varnothing _{\phi }$$  $$x\Vdash \varphi \wedge \psi$$  iff  $$x\Vdash \varphi$$ and $$x\Vdash \psi$$  $$y\Vdash ^{\!\!\partial }\varphi \vee \psi$$  iff  $$y\Vdash ^{\!\!\partial }\varphi$$ and $$y\Vdash ^{\!\!\partial }\psi$$  $$x\Vdash \varphi \vee \psi$$  iff  $$\forall y\;(xRy\;\Longrightarrow \;y\not \Vdash ^{\!\!\partial }\varphi \ \textrm{or}\ y\not \Vdash ^{\!\!\partial }\psi )$$  $$y\Vdash ^{\!\!\partial }\varphi \wedge \psi$$  iff  $$\forall x\;(xRy\;\Longrightarrow \;x\not \Vdash \varphi \ \textrm{or}\ x\not \Vdash \psi )$$  $$x\Vdash \varphi \circ \psi$$  iff  $$\exists u, z\;(xR_{\circ } uz\ \textrm{and}\ u\Vdash \varphi \ \textrm{and}\ z\Vdash \psi )$$  $$y\Vdash ^{\!\!\partial }\varphi \circ \psi$$  iff  $$\forall x,z\;(x\Vdash \varphi \ \textrm{and}\ z\Vdash \psi \;\Longrightarrow \; yR^{\partial }_{\circ } xz)$$  $$x\Vdash \varphi \rightarrow \psi$$  iff  $$\forall z\forall y\;(z\Vdash \varphi \ \textrm{and}\ y\Vdash ^{\!\!\partial }\psi \;\Longrightarrow \;xS_>zy)$$  $$y\Vdash ^{\!\!\partial }\varphi \rightarrow \psi$$  iff  $$\exists x, v\;(yR^{\partial }_>xv\ \textrm{and}\ x\Vdash \varphi \ \textrm{but}\ v\Vdash ^{\!\!\partial }\psi )$$  (x, u, z ∈ X, y, v ∈ Y, R = $${\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}$$?, $$\varnothing _{\psi }=\bigcap{\mathcal G}_{\psi }(X),\;\varnothing _{\phi }=\bigcap{\mathcal G}_{\phi }(Y)$$)  $$x\Vdash p$$  iff  x ∈ V (p)  $$y\Vdash ^{\!\!\partial } p$$  iff  $$y\in V{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}(p)$$  $$x\Vdash \top$$    always  $$y\Vdash ^{\!\!\partial }\bot$$    always  $$x\Vdash \bot$$  iff  $$x\in \varnothing _{\psi }$$  $$y\Vdash ^{\!\!\partial }\top$$  iff  $$y\in \varnothing _{\phi }$$  $$x\Vdash \varphi \wedge \psi$$  iff  $$x\Vdash \varphi$$ and $$x\Vdash \psi$$  $$y\Vdash ^{\!\!\partial }\varphi \vee \psi$$  iff  $$y\Vdash ^{\!\!\partial }\varphi$$ and $$y\Vdash ^{\!\!\partial }\psi$$  $$x\Vdash \varphi \vee \psi$$  iff  $$\forall y\;(xRy\;\Longrightarrow \;y\not \Vdash ^{\!\!\partial }\varphi \ \textrm{or}\ y\not \Vdash ^{\!\!\partial }\psi )$$  $$y\Vdash ^{\!\!\partial }\varphi \wedge \psi$$  iff  $$\forall x\;(xRy\;\Longrightarrow \;x\not \Vdash \varphi \ \textrm{or}\ x\not \Vdash \psi )$$  $$x\Vdash \varphi \circ \psi$$  iff  $$\exists u, z\;(xR_{\circ } uz\ \textrm{and}\ u\Vdash \varphi \ \textrm{and}\ z\Vdash \psi )$$  $$y\Vdash ^{\!\!\partial }\varphi \circ \psi$$  iff  $$\forall x,z\;(x\Vdash \varphi \ \textrm{and}\ z\Vdash \psi \;\Longrightarrow \; yR^{\partial }_{\circ } xz)$$  $$x\Vdash \varphi \rightarrow \psi$$  iff  $$\forall z\forall y\;(z\Vdash \varphi \ \textrm{and}\ y\Vdash ^{\!\!\partial }\psi \;\Longrightarrow \;xS_>zy)$$  $$y\Vdash ^{\!\!\partial }\varphi \rightarrow \psi$$  iff  $$\exists x, v\;(yR^{\partial }_>xv\ \textrm{and}\ x\Vdash \varphi \ \textrm{but}\ v\Vdash ^{\!\!\partial }\psi )$$  Proposition 5.6 (Soundness) Full BCK is sound in the class of general frames defined. Proof. Letting $$[\![\varphi ]\!]=\{x\in X\;|\; x\Vdash \varphi \}$$ and $$(\!|\varphi |\!) =\{y\in Y\;|\; y\Vdash ^{\!\!\partial }\varphi \}$$, the reader can easily verify that conjunctions and disjunctions are interpreted as intersections and closures of unions, respectively, while $$[\![\varphi \circ \psi ]\!]=[\![\varphi ]\!]\ {\bigcirc}{\kern-6.4pt\mid}\ \,\ [\![\psi ]\!]$$, $$[\![\varphi \rightarrow \psi ]\!]=[\![\varphi ]\!]\Rightarrow [\![\psi ]\!]$$ and, dually, $$(\!|\varphi \circ \psi |\!) =(\!|\varphi |\!)\ {\bigcirc}{\kern-6.4pt\mid}\ \,^{\partial }(\!|\psi |\!)$$ and $$(\!|\varphi \rightarrow \psi |\!) =(\!|\varphi |\!) \Rightarrow ^{\partial }(\!|\psi |\!)$$. This follows from the fact that the restriction that $$V{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}(p)=V(p){ }^{{\mathop{=}^{\kern-5pt\shortmid}}}$$ effectively imposes that propositional variables are (co)interpreted as clopen sets. Soundness then follows by Lemmas 5.3, 5.4. Theorem 5.7 (Completeness) Full BCK is complete in the class of general frames defined. Proof. Completeness follows immediately by the representation argument presented in Section 4, with arguments instantiated to the distribution types of interest for BCK. The canonical representation map indeed sends the equivalence class $$a=[\vartheta]$$ of a sentence $$\vartheta$$ to the clopen set $$\Gamma x_{a}=\{x\in X\;|\; a\in x\}$$ and, similarly for the co-representation map. Details can be safely left to the interested reader. 6 Conclusions We have shown in this article that the duality of the categories of bounded lattices and ⊥-frames proven in [19] lifts to a duality of the categories of normal lattice expan-sions of similarity type $$\tau$$ with the corresponding categories of $$\tau$$-frames. A lattice operator, of some distribution type $$\delta$$, is represented as a set operator $${\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta }$$ or $$\ominus _{\delta }$$, depending on the output type of $$\delta$$, canonically generated by a relation, while the restriction of $${\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta }/\ominus _{\delta }$$ on closed sets (of its domain) is the $$\sigma$$-extension of its restriction on clopens, extended to all (co)stable sets using join-density of closed sets. Extensions of normal lattice operators in canonical extensions have been algebraically constructed in [9] and, in the context of a Stone type duality, in [24, 25]. The contribution of the present article lies in the fact that it presents a relational representation of the operators which can be applied to provide relational (Kripke-style) semantics to various logical calculi lacking the axiom of distribution of conjunction over disjunction and conversely. The present article constitutes an abstraction and generalisation over [20] (http://rgdoi.net/10.13140/RG.2.2.34134.55362/1), an article by this author treating the semantics of modal logic over an implicative, non-distributive lattice with an intuitionistic type of negation. A first version of the current article was made public under the title ‘Relational representation of operators in canonical extensions of normal lattice expansions’ [21]. To illustrate the approach and keep this article self-contained, we have included an application in Section 5, modeling full BCK. In as far as the semantics of non-distributive logical calculi is concerned we repeat here our conclusion from [20] that there appears to exist a choice to be made, namely between pursuing a uniform algebraic approach based on canonical extensions and then abandoning the standard interpretation of e.g. boxes and diamonds, or taking a more applied stance and preferring to abandon uniformity of approach when it comes to semantic issues. The choice boils down to either (a) considering all interpretations assigning just any stable set to propositional variables as admissible, but then the received interpretation of familiar operators must be abandoned, or (b) we may opt for recapturing the familiar meaning of operators despite the absence of distribution, but then interpretations must be restricted to the closed ones, assigning a closed element to a propositional variable. The present article presents the necessary duality theory in support of either one of the above choices. References [1] G. Allwein and C. Hartonas. Duality for bounded lattices. Technical Report IULG-93–25. Indiana University Logic Group, 1993. [2] K. Bimbó. Functorial duality for ortholattices and de morgan lattices. Logica Universalis , 1, 311– 333, 2007. Google Scholar CrossRef Search ADS   [3] G. Birkhoff. Lattice Theory . American Mathematical Society Colloquium Publications 25 , 3rd edn. American Mathematical Society, Providence, Rhode Island, 1979 (corrected reprint of the 1967 3rd edn.). [4] W. Conradie and A. Palmigiano. Algorithmic correspondence and canonicity for non-distributive logics, 2016. arXiv:1603.08515v2. [5] A. Craig, M. J. Gouveia and M. Haviar. TiRS graphs and TiRS frames: a new setting for duals of canonical extensions. Algerba Universalis , 74, 2015. [6] J. M. Dunn. Gaggle theory: an abstraction of galois coonections and resuduation with applications to negations and various logical operations. In Logics in AI, Proceedings of European Workshop JELIA 1990, LNCS 478 , pp. 31– 51, 1990. [7] I. Düntsch, E. Orlowska, A. M. Radzikowska and D. Vakarelov. Relational representation theorems for some lattice-based structures. Journal of Relational Methods in Computer Science (JORMICS) , 1, 132– 160, 2004. [8] M. Gehrke. Generalized Kripke frames. Studia Logica , 84, 241– 275, 2006. Google Scholar CrossRef Search ADS   [9] M. Gehrke and J. Harding. Bounded lattice expansions. Journal of Algebra , 238, 345– 371, 2001. Google Scholar CrossRef Search ADS   [10] M. Gehrke and B. Jónsson. Bounded distributive lattice expansions. Math. Scand. , 94, 13– 45, 2004. Google Scholar CrossRef Search ADS   [11] C. Hartonas. Order-duality, negation and lattice representation. In Negation: A Notion in Focus, H. Wansing ed., pp. 27– 36. de Gruyter, 1996. [12] C. Hartonas. Duality for lattice-ordered algebras and for normal algebraizable logics. Studia Logica , 58, 403– 450, 1997. Google Scholar CrossRef Search ADS   [13] C. Hartonas. First-order frames for orthomodular quantum logic. Journal of Applied Non-Classical Logics , 26, 69– 80, 2016. Google Scholar CrossRef Search ADS   [14] C. Hartonas. Order-dual relational semantics for non-distributive propositional logics: a general framework. Journal of Philosophical Logic , 47, 67– 94, 2018. Google Scholar CrossRef Search ADS   [15] C. Hartonas. Reasoning with incomplete information in generalized Galois logics without distribution: the case of negation and modal operators. In J. M. Dunn on Information Based Logics, K. Bimbó, ed., pp. 303– 336, Springer series Outstanding Contributions to Logic, 2016. [16] C. Hartonas. Kripke-Galois frames and their logics. IFCoLog Journal of Logics and their Applications , 2017. [17] C. Hartonas. Order-dual relational semantics for non-distributive propositional logics. Oxford Logic Journal of the IGPL , 25, 145– 182, 2017. [18] C. Hartonas and J. M. Dunn. Duality theorems for partial orders, semilattices, galois connections and lattices. Technical Report IULG-93–26. Indiana University Logic Group, 1993. [19] C. Hartonas and J. M. Dunn. Stone duality for lattices. Algebra Universalis , 37, 391– 401, 1997. Google Scholar CrossRef Search ADS   [20] C. Hartonas. Canonical extensions and Kripke-Galois semantics for non-distributive propositional logics, 2017. [21] C. Hartonas. Relational representation of operators in canonical extensions of normal lattice expansions, 2017. [22] G. Hartung. A topological representation for lattices. Algebra Universalis , 29, 273– 299, 1992. Google Scholar CrossRef Search ADS   [23] B. Jónsson and A. Tarski. Boolean algebras with operators I, II. Americal Journal of Mathematics , 71, 109– 126, 2014. [24] M. A. Moshier and P. Jipsen. Topological duality and lattice expansions, i: a topological construction of canonical extensions. Algebra Universalis , 71, 109– 126, 2014. Google Scholar CrossRef Search ADS   [25] M. A. Moshier and P. Jipsen. Topological duality and lattice expansions, ii: lattice expansions with quasioperators. Algebra Universalis , 71, 221– 234, 2014. Google Scholar CrossRef Search ADS   [26] H. Ono. Structural rules and a logical hierarchy. In Proceedings of the Summer School and Conference on Mathematical Logic (Chaika, Bulgaria, 1988), P. P. Petkov ed., pp. 95– 104. Plenum Press, New York, 1990. [27] H. Ono. Algebraic aspects of logics without structural rules. AMS Contemporary Mathematics , 131, 601– 621, 1992. Google Scholar CrossRef Search ADS   [28] H. Ono. Logics without the contraction rule and residuated lattices. Australasian Journal of Logic , 8, 50– 81, 2010. [29] H. Ono and Y. Komori. Logics without the contraction rule. The Journal of Symbolic Logic , 50, 169– 201, 1985. Google Scholar CrossRef Search ADS   [30] M. Ploščica. A natural representation of bounded lattices. Tatra Mountains Math. Publ. , 5, 75– 88, 1995. [31] H. Priestley. Representation of distributive lattices by means of ordered stone spaces. Bull. Lond. Math. Soc. , 2, 186– 190, 1970. Google Scholar CrossRef Search ADS   [32] M. H. Stone. Topological representation of distributive lattices and brouwerian logics. Casopsis pro Pestovani Matematiky a Fysiky , 67, 1– 25, 1937. [33] M. H. Stone. The representation of boolean algebras. Bull. Amer. Math. Soc. , 44, 807– 816, 1938. Google Scholar CrossRef Search ADS   [34] T. Suzuki. Bi-approximation semantics for substructural logic at work. Advances in Modal Logic , 8, 411– 433, 2010. [35] A. Urquhart. A topological representation of lattices. Algebra Universalis , 8, 45– 58, 1978. Google Scholar CrossRef Search ADS   [36] A. Urquhart. Duality for algebras of relevant logics. Studia Logica: An International Journal for Symbolic Logic , 56, 263– 276, 1996. © The Author(s) 2018. Published by Oxford University Press. All rights reserved. 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# Stone duality for lattice expansions

, Volume Advance Article – Apr 26, 2018
30 pages

/lp/ou_press/stone-duality-for-lattice-expansions-BgGf9QvNKz
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Oxford University Press
ISSN
1367-0751
eISSN
1368-9894
D.O.I.
10.1093/jigpal/jzy010
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### Abstract

Abstract The Stone duality for bounded lattices by this author, with J.M. Dunn, is lifted in this article to a duality for lattices with operators. The dual frames of lattice expansions are two-sorted frames (X, $${\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}$$, Y), further equipped with an (n + 1)-ary relation $$\textit{R}_{\delta }$$ and a dual relation $$\textit{R}^{\partial }_{\delta }$$ for each n-ary lattice operator of some distribution type $$\delta$$. The closures of the generalized image operators generated by the relations are shown to be precisely the $$\sigma$$-extensions of the corresponding lattice operators and thus the full complex algebra of Galois-stable sets of the frame constitutes a concrete canonical extension of the lattice expansion. Thereby, the results presented in this article extend to the non-distributive case the classical Jónsson–Tarski results for Boolean algebras with operators and their extension to mere distributive lattices with operators. Consequently, the duality based approach to relational logic semantics is extended here to the case of logics dropping distribution. As an application example, we model the full BCK calculus. Both plain Kripke-type (two-sorted) frame semantics, as well as general (two-sorted) frame semantics are presented, the distinctive feature of the latter choice being that the interpretation of additional lattice operators (such as modal operators) is typically verbatim the same as in the distributive case, which is desirable in intended applications (such as temporal, or dynamic extensions of non-distributive lattice logic). 1 Introduction This article lies in the tradition of Stone dualities for lattice-based algebras [31–33] and its related set-theoretic (relational) semantics tradition for the associated logical calculi, initiated by the classical Jónsson–Tarski results on Boolean algebras with operators [23] and extended to the case of a mere distributive lattice by Urquhart [36] and others. The article builds on the theory of canonical extensions [9, 10] and on this author’s work, both past [11, 12, 19] and recent [13–17] and it constitutes a generalisation and abstraction over a recent article [20]. In [24, 25] Moshier and Jipsen studied a representation of lattices with operators, extended to a duality. Unlike [24, 25], the focus of this article is on the relational representation of normal operators on bounded lattices. The representation is extended to a full functorial duality, obtained by lifting an existing duality [19] by this author, with J.M. Dunn, for the category of bounded lattices. In [14], the same objectives were addressed, based on a duality for lattices with additional operators [12] and resulting in the development of the framework of order-dual, or Kripke–Galois semantics for non-distributive logics. We take as our starting point here the lattice representation [19], which constitutes a canonical extension of the lattice, as shown in [9], Proposition 2.6. In as far as applications in logic are concerned, our results deliver a 2-sorted version of the Kripke–Galois semantics framework of [2, 14]. The semantics of non-distributive logics has been also recently addressed by various other authors [4, 5, 7, 8, 34], invariably leading to non-standard interpretations of familiar logical operators (such as diamonds and boxes), which appears to be necessary when working with plain Kripke-type frame semantics. Our results serve as a basis both for plain Kripke-type frame semantics, as well as for a semantics based on general frames. In the latter case, the received (intended) interpretation of operators is recaptured, which is desirable in some applied contexts, such as temporal, or dynamic extensions of non-distributive lattice logic. The present article is also related to and it constitutes an advancement of Dunn’s theory of generalized Galois logics [6] and it sheds light on the semantics of substructural logics, specifically for the case where distribution of conjunction over disjunction and conversely is not assumed in the logic. For the reader’s convenience, we gather some technical preliminaries in Section 2. Specifically, Section 2.4 recalls the basic notions from the theory of canonical extensions [9, 10] and it gives a brief presentation of the lattice representation of [19]. Section 2.5 recalls the construction in a canonical lattice extension of the $$\sigma /\pi$$-extensions of lattice operators [9]. Section 2.1 presents the elementary properties of polarities (X, $${\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}$$, Y) that will be useful in the sequel, where X, Y are nonempty sets and $${\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}$$ ⊆ X × Y. Section 2.3 defines the category L$$_{\tau }$$ of normal lattice expansions of some similarity type $$\tau$$. Section 3 defines the category of $$\tau$$-frames, in steps. First, in Section 3.1 we consider base frames (polarities) with relations, we define the generated generalized image operators and we prove that they restrict to operators on the families of closed and of clopen sets of their respective domains and that their dual operators can be obtained from the duals of the frame relations, which we define in a canonical way. Section 2.2 reviews the definition of lattice frames and their morphisms, from the lattice representation of [19]. It is in Section 3.2 that objects of the category of $$\tau$$-frames are axiomatized and their morphisms are defined. Finally, Section 4 is devoted to the proof that the duality of the categories of bounded lattices and ⊥-frames proven in [19] lifts to a duality of the categories of normal lattice expansions of similarity type $$\tau$$ with the corresponding categories of $$\tau$$-frames. Much of Section 4 is devoted to the construction of the dual frame of a lattice expansion $${\mathcal A}_{\tau }=({\mathcal L},(\,f_{\delta })_{\delta \in \tau _{1}},(h_{\delta })_{\delta \in \tau _{\partial }})$$. To define the canonical relations on the frame we make use of the point operators we introduced in [12]. The key point in the argument is the proof that the restriction of a generated set-operator on the closed sets of its domain is the $$\sigma$$-extension of its restriction to clopen sets. Full functorial duality is proven in Theorem 4.5. Section 5 presents an application of the developed framework, modeling full BCK. For the reader’s benefit, we present both a plain Kripke-type semantics for BCK, as well as a semantics based on general frames, both using the representation results of this article. 2 Technical preliminaries 2.1 Polarities A polarity (base frame) is a triple (X, $${\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}$$, Y) where X, Y are nonempty sets (of worlds and co-worlds) and $${\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}$$ ⊆ X × Y is a binary relation, to be called the Galois relation of the frame, generating a Galois connection defined [3] on U ⊆ X and V ⊆ Y by   \begin{align*} \phi(U)=U^{{\mathop{=}^{\kern-5pt\shortmid}}} =&\{y\in Y\;|\;\forall u\in U\; u\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; y\}=\{y\in Y\;|\; U\ {\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\ y\}\\ \psi(V)={}^{{\mathop{=}^{\kern-5pt\shortmid}}} V =&\{x\in X\;|\; \forall v\in V\; x\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; v\}=\{x\in X\;|\; x\ {\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\ V\}. \end{align*}A subset A ⊆ X is Galois-stable if $$A=\psi \phi (A)$$ and we let $${\mathcal G}_{\psi }(X)$$ be the complete lattice of Galois-stable subsets of X. Similarly for $${\mathcal G}_{\phi }(Y)$$ and the complete lattice of co-stable subsets of Y, $$B=\phi \psi (B)$$. We also let $$\varnothing _{\psi },\varnothing _{\phi }$$ be the least elements of $${\mathcal G}_{\psi }(X)$$ and $${\mathcal G}_{\phi }(Y)$$, respectively, i.e. the intersections of all their members, and we note that they need not be empty. The relations x ≤ z iff $$\{x\}^{{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}} }\subseteq \{z\}^{{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}} }$$ on X and y ≤ v iff $${ }^{{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}} }\{y\}\subseteq{ }^{{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}} }\{z\}$$ on Y are preorders on X and Y, respectively. We make the further assumptions that (F1) X, Y are both partial orders under their respective ≤-relation (F2) $${\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}$$ is increasing in each argument place, i.e. x$${\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}$$y, x ≤ z, y ≤ v imply z$${\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}$$v. Note that the partial-ordering assumption implies that base frames are separated (S-frames), in the sense of [8]. We use $$\Gamma$$ for the upper closure operator, $$\Gamma U=U^{\uparrow }$$. In particular, for x ∈ X (resp. y ∈ Y) we write $$\Gamma x$$ for the principal upper set over x, as shorthand for the more accurate $$\Gamma (\{x\})$$: $$\Gamma x=\{z\in X\;|\; x\leq z\}$$. Similarly for $$\Gamma y$$, with y ∈ Y. Lemma 2.1 Let (X, $${\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}$$, Y) be a base frame. Then the following hold, for any x ∈ X, y ∈ Y, (1) $$(\Gamma x){ }^{{\mathop{=}^{\kern-5pt\shortmid}}}=\{x\}{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}$$ and $${ }^{{\mathop{=}^{\kern-5pt\shortmid}}}(\Gamma y)={ }^{{\mathop{=}^{\kern-5pt\shortmid}}}\{y\}$$ (2) $$\Gamma x\in \mathcal{G}_{\psi }(X)$$ and $$\Gamma y\in \mathcal{G}_{\phi }(Y)$$. Hence also $${ }^{{\mathop{=}^{\kern-5pt\shortmid}}}\{y\}\in \mathcal{G}_{\psi }(X)$$ and $$\{x\}{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}\in \mathcal{G}_{\phi }(Y)$$ (3) Stable and co-stable sets are increasing, i.e. u ∈ A implies $$\Gamma u\subseteq A$$ (4) For any $$A\in \mathcal{G}_{\psi }(X)$$, $$A=\bigvee _{x\in A}\Gamma x=\bigcap _{A {\mathop{=}^{\kern-5pt\raise-4pt\shortmid}} y}({ }^{{\mathop{=}^{\kern-5pt\shortmid}}}\{y\})$$ and similarly for $$B\in{\mathcal G}_{\phi }(Y)$$. Proof. For 1), left-to-right is immediate and the other direction uses increasingness of $${\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}$$. For 2), use 1) and the definition of the partial order. For 3) we do only the case for stable sets, since the argument for co-stable sets is completely similar. We have $$x\in A\Longrightarrow \{x\}\subseteq A\Longrightarrow A{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}\subseteq \{x\}{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}=(\Gamma x){ }^{{\mathop{=}^{\kern-5pt\shortmid}}}$$ , using claim 1). Given that $$\Gamma x$$ is stable, by claim 2), we obtain $$\Gamma x={ }^{{\mathop{=}^{\kern-5pt\shortmid}}}((\Gamma x){ }^{{\mathop{=}^{\kern-5pt\shortmid}}})\subseteq{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}(A{ }^{{\mathop{=}^{\kern-5pt\shortmid}}})=A$$. 4) now follows, since clearly $$A\subseteq \bigvee _{x\in A}\Gamma x$$ and, conversely, $$x\in A\Longrightarrow \Gamma x\subseteq A$$, using claim 3), hence $$\bigvee _{x\in A}\Gamma x\subseteq A$$, as well. Similarly for $$B\in{\mathcal G}_{\phi }(Y), B=\bigvee _{y\in B}\Gamma y$$. Then $$A{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}=\bigvee _{y\in A{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}}\Gamma y$$, hence applying $$\psi$$ we obtain $$A=\bigcap _{A\subseteq{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}\{y\}}({}^{{\mathop{=}^{\kern-5pt\shortmid}}}\{y\})=\bigcap _{A\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; y}({ }^{{\mathop{=}^{\kern-5pt\shortmid}}}\{y\})$$. Let $${\mathcal G}_{\kappa }(X)=\{\Gamma x\;|\; x\in X\}$$ and $${\mathcal G}_{o}(X)=\{{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}\{y\}\;|\; y\in Y\}$$ and similarly for $${\mathcal G}_{\kappa }(Y),{\mathcal G}_{o}(Y)$$. It has been shown above that Corollary 2.2 $${\mathcal G}_{\kappa }(X)$$ is join-dense in $${\mathcal G}_{\psi }(X)$$, while $${\mathcal G}_{o}(X)$$ is meet-dense in $${\mathcal G}_{\psi }(X)$$ and similarly for $${\mathcal G}_{\phi }(Y)$$. Definition 2.3 (Closed and open elements) The closed (filter) elements of $${\mathcal G}_{\psi }(X)$$ are the join-generators $$\Gamma x$$, with x ∈ X, and its open (ideal) elements are the meet-generators $${ }^{{\mathop{=}^{\kern-5pt\shortmid}}}\{y\}$$, with y ∈ Y. Similarly for $${\mathcal G}_{\phi }(Y)$$. A stable set A is a clopen element of $${\mathcal G}_{\psi }(X)$$ if it is both closed and open, i.e. $$\Gamma x=A={ }^{{\mathop{=}^{\kern-5pt\shortmid}}}\{y\}$$, for some x ∈ X, y ∈ Y, both necessarily unique. Similarly for clopen elements $$\{x\}{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}=B=\Gamma y$$ of $${\mathcal G}_{\phi }(Y)$$. We let $${\mathcal G}_{\kappa o}$$ designate clopen elements. By a slight abuse of terminology, a point x ∈ X will be called clopen when $$\Gamma x$$ is clopen and similarly for y ∈ Y. Note that, since reversing the order switches meets and joins, the closed elements of $${\mathcal G}_{\psi }(X)^{\partial }$$ are the ones that are open in $${\mathcal G}_{\psi }(X)$$, i.e. the elements $${ }^{{\mathop{=}^{\kern-5pt\shortmid}}}\{y\}$$, while its open elements are, dually, the closed elements of $${\mathcal G}_{\psi }(X)$$. Remark 2.4 (Closed elements of products) Up to isomorphism, the closed elements of a product are tuples of closed elements of the factors of the product and similarly for open elements. Since we shall often have use of closed and open elements of products, it is useful to point out that in e.g. $${\mathcal G}_{\psi }(X)\times{\mathcal G}_{\psi }(X)^{\partial }$$ a closed element is a pair $$(K,K^{\prime })$$ where K is closed in $${\mathcal G}_{\psi }(X)$$ and $$K^{\prime }$$ is closed in $${\mathcal G}_{\psi }(X)^{\partial }$$, hence open in $${\mathcal G}_{\psi }(X)$$. It is therefore a pair of the form $$(\Gamma x,{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}\{y\})$$. More generally, if $$\delta =(i_{1},\ldots ,i_{n};i_{n+1})$$ is a distribution type of a normal lattice operator f, we will be constructing maps $$F:{\mathcal G}_{\psi }(X)^{i_{1}}\times \cdots \times{\mathcal G}_{\psi }(X)^{i_{n}}\longrightarrow{\mathcal G}_{\psi }(X)^{i_{n+1}}$$, defined on closed elements of the product, i.e. tuples of the form $$(\ldots ,\underbrace{\Gamma u_{j}}_{i_{j}=1},\ldots ,\underbrace{{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}\{u_{r}\}}_{i_{r}=\partial },\ldots )$$, where for $$i_{j}=1,\; u_{j}\in X$$, while for $$i_{r}=\partial ,\;u_{r}\in Y$$. Dually, an open element of the product is a tuple of the form $$(\ldots ,\underbrace{{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}\{ u_{j}\}}_{i_{j}=1},\ldots ,\underbrace{\Gamma u_{r}}_{i_{r}=\partial },\ldots )$$, where for $$i_{j}=1,\; u_{j}\in Y$$, while for $$i_{r}=\partial ,\;u_{r}\in X$$. The Galois maps $$\psi ={ }^{{\mathop{=}^{\kern-5pt\shortmid}}}(\;),\phi =(\;){ }^{{\mathop{=}^{\kern-5pt\shortmid}}}$$ provide a dual equivalence $${\mathcal G}_{\psi }(X)^{\partial }\backsimeq{\mathcal G}_{\phi }(Y)$$. It is also useful to work with two-sorted products whose factors are either $${\mathcal G}_{\psi }(X)$$ or $${\mathcal G}_{\phi }(Y)$$. If $$\delta =(i_{1},\ldots ,i_{n};i_{n+1})$$, we shall also be considering products $${\mathcal G}(Z^{i_{1}})\times \cdots \times{\mathcal G}(Z^{i_{n}})\longrightarrow{\mathcal G}(Z^{i_{n+1}})$$, where $${\mathcal G}(Z^{i_{j}})$$ is $${\mathcal G}_{\psi }(X)$$ if $$i_{j}=1$$ and it is $${\mathcal G}_{\phi }(Y)$$ if $$i_{j}=\partial$$. Closed elements of produts are then tuples $$(\Gamma u_{1},\ldots ,\Gamma u_{n})$$, where $$\Gamma u_{j}\in{\mathcal G}_{\psi }(X)$$ (for some $$u_{j}\in X$$) if $$i_{j}=1$$ and $$\Gamma u_{r}\in{\mathcal G}_{\phi }(Y)$$ (for some $$u_{r}\in Y$$) if $$i_{r}=\partial$$. 2.2 Lattice frames Lattice frames (X, $${\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}$$, Y) have been defined in [19], involving topologies on each of X, Y. We will therefore avoid re-inventing the wheel and we shall base our definition on pre-existing work. For a lattice frame we require in addition to axioms (F1,F2) for base frames that (F3) Clopen sets are closed under finite intersections and closed sets are closed under arbitrary intersections, for each of $${\mathcal G}_{\psi }(X),{\mathcal G}_{\phi }(Y)$$. (F4) If A is an index set for clopen elements $$\Gamma x_{a}={ }^{{\mathop{=}^{\kern-5pt\shortmid}}}\{y_{a}\}$$, the set $$\{\Gamma x_{a}\;|\; a\in A\}\cup \{-\Gamma x_{a}\;|\; a\in A\}$$ is a subbasis for a compact, totally separated topology on X. Similarly for Y. In other words, X, Y are both Stone spaces. (F5) The family of closed sets is the meet-closure of the family of clopen sets, for each of $${\mathcal G}_{\psi }(X),{\mathcal G}_{\phi }(Y)$$. Axioms (F3–F5) are equivalent to the conditions defining FSpaces in [19]. By choice of the subbasis, clopens in the algebraic sense are precisely the clopen sets in the topological sense. The axioms clearly imply that clopens form a lattice, with joins defined by $$\Gamma x_{a}\vee \Gamma x_{b}=\psi (\phi \Gamma x_{a}\cap \phi \Gamma x_{b})=\psi (\{x_{a}\}{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}\cap \{x_{b}\}{ }^{{\mathop{=}^{\kern-5pt\shortmid}}})=\psi (\Gamma y_{a}\cap \Gamma y_{b})$$. They also imply that $${\mathcal G}_{\kappa }(X)$$ is a complete meet-subsemillatice of $${\mathcal G}_{\psi }(X)$$, hence a complete lattice (but arbitrary joins do not necessarily coincide in $${\mathcal G}_{\kappa }(X),{\mathcal G}_{\psi }(X)$$). This induces a complete lattice structure on X (and similarly for Y). Furthermore, every closed element $$\Gamma x$$ is obtained as the meet of the clopens that cover it, i.e. $$\Gamma x=\bigwedge \{\Gamma x_{a}\;|\;\Gamma x\subseteq \Gamma x_{a}\}$$ and similarly for $${\mathcal G}_{\kappa }(Y)$$. Consequently, clopens are meet-dense in the lattice of closed elements. Furthermore, meet-density of clopens in $${\mathcal G}_{\phi }(Y)$$ evidently implies that clopens are also join-dense in $${\mathcal G}_{\kappa }(X)$$ (and similarly for $${\mathcal G}_{\kappa }(Y)$$ and $${\mathcal G}_{\phi }(Y)$$). Lemma 2.5 (1) Each of X, Y is an FSpace, in the sense of [19]. (2) The frame (X, $${\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}$$, Y) is a ⊥-frame, in the sense of [19]. (3) $$(\imath ,{\mathcal G}_{\psi }(X))$$ is a canonical extension of the lattice $${\mathcal G}_{\kappa o}(X)$$ of clopen elements of $${\mathcal G}_{\psi }(X)$$, where ı is the inclusion of clopens into $${\mathcal G}_{\psi }(X)$$, and $$(\jmath ,{\mathcal G}_{\phi }(Y))$$ is a dual canonical extension, where $$\jmath (\Gamma x_{a})=\{x_{a}\}{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}=\Gamma y_{a}$$ is the inclusion map of clopens into $${\mathcal G}_{\phi }(Y)$$. Proof. The proof for 1) and 2) is immediate, by consulting the definitions in [19]. For 3), combine with the results of [9], where it is shown that the representation of [19] delivers a canonical extension. Definition 2.6 (Frame morphisms, [19])A morphism$$(\,f,h):(X_{1},{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}} _{1},Y_{1})\longrightarrow (X_{2},{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}} _{2},Y_{2})$$of lattice frames is a pair of functions $$f:X_{1}\longrightarrow X_{2},\;h:Y_{1}\longrightarrow Y_{2}$$, such that f, h are continuous functions that preserve greatest lower bounds in $$X_{1},Y_{1}$$, respectively, and whose inverse image takes clopens to clopens (called FSpace-morphisms in [19]). Setting $$\,f^{\ast }=f^{-1}, h^{\ast }=h^{-1}$$, both squares in the diagram below commute where $$X_{i}^{\ast },Y_{i}^{\ast }$$ stand for the respective collections of clopen sets. 2.3 Normal lattice expansions This section introduces categories of algebraic structures of interest in the present article, i.e. expansions of bounded lattices by normal operators, typically arising as the Lindenbaum–Tarski algebras of logical calculi. By a distribution type we mean an element $$\delta$$ of the set $$\{1,\partial \}^{n+1}$$, for some n ≥ 0, typically to be written as $$\delta =(i_{1},\ldots ,i_{n};i_{n+1})$$ and where $$\delta _{(n+1)}=i_{n+1}\in \{1,\partial \}$$ will be referred to as the output type of $$\delta$$. A similarity type$$\tau$$ is then defined as a finite sequence of distribution types, $$\tau =\langle \delta _{1},\ldots ,\delta _{k}\rangle$$. Definition 2.7 (Normal operators) Following [23], an n-ary monotone operator $$f:{\mathcal L}^{n}\longrightarrow{\mathcal L}$$ will be called additive if it distributes over joins of $$\mathcal L$$ in each argument place. More generally, if $${\mathcal L}_{1},\ldots ,{\mathcal L}_{n},{\mathcal L}$$ are bounded lattices, then a monotone function $$f:{\mathcal L}_{1}\times \cdots \times{\mathcal L}_{n}\longrightarrow{\mathcal L}$$ is additive, if for each i, f distributes over binary joins of $${\mathcal L}_{i}$$, i.e. $$f(a_{1},\ldots ,a_{i-1},b\vee d,a_{i+1},\ldots ,a_{n})=f(a_{1},\ldots ,a_{i-1},b,a_{i+1},\ldots ,a_{n})\vee f(a_{1},\ldots ,a_{i-1},d,a_{i+1},\ldots ,a_{n})$$. As a matter of notation, we write $$\mathcal L$$ for $${\mathcal L}^{1}$$ and $${\mathcal L}^{\partial }$$ for its opposite lattice (where order is reversed, usually designated as $${\mathcal L}^{op}$$). An n-ary operator f on a lattice $$\mathcal L$$ is normal [12] if it is an additive function $$f:{\mathcal L}^{i_{1}}\times \cdots \times{\mathcal L}^{i_{n}}\longrightarrow{\mathcal L}^{i_{n+1}}$$, where each $$i_{j}$$, for $$j=1,\ldots ,n,n+1$$, is in the set {1, ∂}, i.e. $${\mathcal L}^{i_{j}}$$ is either $$\mathcal L$$, or $${\mathcal L}^{\partial }$$. For a normal operator f on $$\mathcal L$$, its distribution type is the $$(n+1)$$-tuple $$\delta (f)=(i_{1},\ldots ,i_{n};i_{n+1})$$. We call f completely normal if it (co)distributes over arbitrary joins, or meets, at each argument place. Definition 2.8 A lattice expansion is a structure $${\mathcal L}=(L,\wedge ,\vee ,0,1,(\,f_{i})_{i\in k})$$ where k > 0 is a natural number and for each i ∈ k, $$\;f_{i}$$ is a normal operator on $$\mathcal L$$ of some specified arity $$\alpha (\,f_{i})\in \mathbb{N}^{+}$$ and distribution type $$\delta (i)$$. The similarity type of $$\mathcal L$$ is the k-tuple $$\tau ({\mathcal L})=\langle \delta (0),\ldots ,\delta (k-1)\rangle$$. Where $$\tau$$ is a similarity type, $$\mathbb{L}_{\tau }$$ is the variety of lattice expansions of similarity type $$\tau$$. L$$_{\tau }$$ designates the category of lattice expansions of similarity type $$\tau$$. Morphisms of lattice expansions are the bound preserving homomorphisms in the usual algebraic sense. Example 2.9 A bounded lattice with a box and a diamond operator $${\mathcal L}=(L,\leq ,\wedge ,\vee ,0,1,\Box ,\Diamond )$$ is an object of L$$_{\tau }$$, where $$\tau$$ is the similarity type $$\tau =\langle (1;1),(\partial ,\partial )\rangle$$ where $$\delta (\Diamond )=(1;1)$$, i.e. $$\Diamond :{\mathcal L}\longrightarrow{\mathcal L}$$ distributes over joins of $$\mathcal L$$, while $$\delta (\Box )=(\partial ;\partial )$$, i.e. $$\Box :{\mathcal L}^{\partial }\longrightarrow{\mathcal L}^{\partial }$$ distributes over ‘joins’ of $${\mathcal L}^{\partial }$$ (i.e. meets of $$\mathcal L$$), delivering ‘joins’ of $${\mathcal L}^{\partial }$$ (i.e. meets of $$\mathcal L$$). An implicative lattice is an object of L$$_{\tau ^{\prime }}$$, where $$\tau ^{\prime }=\langle (1,\partial ;\partial )\rangle$$ and where $$(1,\partial ;\partial )=\delta (\rightarrow )$$ is the distribution type of the implication operator, regarded as a map $$\rightarrow \;:{\mathcal L}\times{\mathcal L}^{\partial }\longrightarrow{\mathcal L}^{\partial }$$ distributing over ‘joins’ in each argument place, i.e. co-distributing over joins in the first place, turning them to meets, and distributing over meets (joins of $${\mathcal L}^{\partial }$$) in the second place, delivering ‘joins’ of $${\mathcal L}^{\partial }$$, i.e. meets of $$\mathcal L$$. An FL-algebra (Full Lambek algebra [27]) is an object of L$$_{\tau ^{^{\prime \prime }}}$$, where $$\tau ^{^{\prime \prime }}=\langle (1,1;1),(1,\partial ;\partial ),(\partial ,1;\partial )\rangle$$, i.e. an algebra $${\mathcal L}=(L,\leq ,\wedge ,\vee ,0,1,\leftarrow ,\circ ,\rightarrow )$$, with $$\delta (\leftarrow )=(\partial ,1;\partial ), \delta (\circ )=(1,1;1)$$ and $$\delta (\rightarrow )=(1,\partial ;\partial )$$. Definition 2.10 A canonical extension of a lattice expansion$$(L,\wedge ,\vee ,0,1,(\,f_{i})_{i\,\in\, k})$$ is a canonical lattice extension $$(\alpha ,C)$$ [9] (see Section 2.4 for a brief review) for the underlying bounded lattice together with an n-ary operator $$F_{i}$$, corresponding to the lattice operator $$f_{i}$$ such that in each argument place if $$f_{i}$$ (co)distributes over finite joins (or meets), then $$F_{i}$$ (co)distributes over arbitrary joins (resp. meets). It is shown in [9] that canonical extensions of normal lattice expansions exist, by constructing extensions $$f_{\sigma }=f_{\pi }$$ (identity follows from the normality assumption for f), as detailed in Section 2.5. 2.4 Canonical extensions For the reader’s convenience, we briefly review in this and the next section the basics on canonical extensions of bounded lattices and $$\sigma /\pi$$-extensions of maps. In [9] a notion of canonical extension of bounded lattices was introduced, generalising the corresponding notion for distributive lattices and Boolean algebras [10] and which characterizes the dual objects of lattices in purely lattice-theoretic terms, without resorting to topological properties. A canonical extension of a bounded lattice $$\mathcal L$$ is defined in [9] as a pair $$(\alpha ,C)$$, where C is a complete lattice and $$\alpha :{\mathcal L}\hookrightarrow C$$ is a lattice embedding and where (density) $$\alpha [{\mathcal L}]$$ is dense in C, where the latter means that every element of C can be expressed both as a meet of joins and as a join of meets of elements in $$\alpha [{\mathcal L}]$$ (compactness) for any set A of closed elements and any set B of open elements of C, $$\bigwedge A\leq \bigvee B$$ iff there exist finite subcollections $$A^{\prime }\subseteq A, B^{\prime }\subseteq B$$ such that $$\bigwedge A^{\prime }\leq \bigvee B^{\prime }$$ where the closed elements of C are defined in [9] as the elements in the meet-closure of the representation map $$\alpha$$ and the open elements of C are defined dually as the join-closure of the image of $$\alpha$$. In other words, if $$\alpha$$ is the representation map and $$M\subseteq{\mathcal L}$$, then $$\bigwedge \{\alpha (a)\;|\; a\in M\}$$ defines a closed element, while $$\bigvee \{\alpha (a)\;|\; a\in M\}$$ defines, dually, an open element. In [9], Proposition 2.6, existence of canonical extensions for bounded lattices is proven by showing that the completion of a bounded lattice $$\mathcal L$$ obtained in the lattice representation theorem of [18, 19] by this author and J.M. Dunn is a canonical extension of $$\mathcal L$$. Furthermore, canonical extensions are proven to be unique, up to isomorphism ([9], Proposition 2.7). Urquhart’s [35] and, subsequently, Hartung’s [22] lattice representations (both predating [18, 19]) also constitute canonical extensions of the represented lattice (see [9], though no proof is presented), as do the representations due to Ploščica [30] and to Allwein and Hartonas [1]. Using Urquhart’s representation, in particular, it is easily proven (see [35]) that the representation reduces to the Priestley representation [31] when the represented lattice is distributive. Our approach in the present article is based on the representation [18, 19] by this author, with J.M. Dunn. In [18] representation and duality theorems for partial orders, semilattices and Galois connections were proven and the lattice representation and duality of [19] is obtained by representing the trivial duality $${\mathcal L}\backsimeq ({\mathcal L}^{\partial })^{\partial }$$, via the identity map. If $${\mathcal L}$$ is a bounded lattice then $${\mathcal L}_{+}=\mathfrak{F}=(X,{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}} , Y)$$, following [19], is the canonical dual frame of the lattice $${\mathcal L}$$, where $$X=\textrm{Filt}({\mathcal L})$$ is the set of lattice filters, $$Y=\textrm{Idl}({\mathcal L})$$ is the set of lattice ideals and where $${\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}$$ ⊆ X × Y represents the trivial (identity) duality $${\mathcal L}\backsimeq ({\mathcal L}^{\partial })^{\partial }$$, explicitly defined by x$${\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}$$y iff ∃a ∈ x i(a) ∈ y iff $$x\cap y\neq \emptyset$$. The relation $${\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}$$, which we refer to as the Galois relation of the frame, generates a Galois connection [3]. It is proven in [9] that $${\mathcal L}^{\sigma }=\{A\subseteq X\;|\; A=\psi \phi (A)\}={\mathcal G}_{\psi }(X)$$ is a canonical extension of $$\mathcal L$$ (dually isomorphic to $$({\mathcal L}^{\sigma })^{\partial }=\{B\subseteq Y\;|\; B=\phi \psi (B)\}={\mathcal G}_{\phi }(Y)$$). The canonical (co)representation maps are given by $$\alpha _{X}(a)=\alpha (a)=\{x\in X\;|\; a\in x\}$$ and $$\alpha _{Y}(a)=\beta (a)=\{y\in Y\;|\; a\in y\}$$. 2.5 $$\sigma ,\pi$$-Extensions of lattice maps If $$(\alpha ,C)$$ is a canonical extension of a bounded lattice $$\mathcal L$$, and K, O are its sets of closed and open elements, the $$\sigma$$ and $$\pi$$-extensions $$f_{\sigma },f_{\pi }:{\mathcal L}_{\sigma }\longrightarrow{\mathcal L}_{\sigma }$$ (where, following the notation of [9], $${\mathcal L}_{\sigma }={\mathcal G}_{\psi }(X)$$ designates the canonical extension of $$\mathcal L$$) of a unary monotone map $$f:{\mathcal L}\longrightarrow{\mathcal L}$$ are defined in [9], taking also into consideration Lemma 4.3 of [9], by setting, for k ∈ K, o ∈ O and u ∈ C  \begin{align} f_{\sigma}(k)=\bigwedge\{\,f(a)\;|\; k\leq a\in L\} \qquad f_{\sigma}(u)=\bigvee\{\,f_{\sigma}(k)\;|\;{\tt K}\ni k\leq u\} \end{align} (1)  \begin{align} f_{\pi}(o)=\bigvee \{\,f(a)\;|\; L\ni a\leq o\} \qquad f_{\pi}(u)=\bigwedge\{\,f_{\pi}(o)\;|\; u\leq o\in{\tt O}\} \end{align} (2)where in these definitions $$\mathcal L$$ is identified with its isomorphic image in C and $$a\in{\mathcal L}$$ is then identified with its representation image. Remark 2.11 In [9], the authors focus on the canonical extension $${\mathcal L}^{\sigma }={\mathcal G}_{\psi }(X)$$ of $$\mathcal L$$ and the use of its dual $${\mathcal G}_{\phi }(Y)\backsimeq ({\mathcal L}^{\sigma })^{\partial }$$ is secondary, since $$({\mathcal L}^{\sigma })^{\partial }\backsimeq ({\mathcal L}^{\partial })^{\sigma }$$ and therefore $${\mathcal G}_{\phi }(Y)$$ is just the canonical extension of $${\mathcal L}^{\partial }$$. Hence they have no use of the dual $$\sigma$$-extensions of maps and they prefer to work with their images in $${\mathcal L}^{\sigma }$$, the $$\pi$$-extensions, obtained by conjugating with the dual equivalence $$\phi \!:{\mathcal G}_{\psi }(X)\!\longleftrightarrow\! {\mathcal G}_{\phi }(Y)^{\partial}\!:\psi$$. It is advantageous for our purposes to work simultaneously with both $${\mathcal G}_{\phi }(Y)$$ and $${\mathcal G}_{\psi }(X)$$ and we will systematically do so. For antitone maps, since the filters of $${\mathcal L}^{\partial }$$ are the ideals of $${\mathcal L}$$, i.e. $$\textrm{Filt}({\mathcal L}^{\partial })=\textrm{Idl}({\mathcal L})$$, and conversely $$\textrm{Idl}({\mathcal L}^{\partial })=\textrm{Filt}({\mathcal L})$$, the canonical frame for $${\mathcal L}^{\partial }$$, after [19], is the frame $$(X^{\prime },{{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}}^{\prime },Y^{\prime })=(Y,{{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}}^{-1},X)$$, where X is the set of filters of $${\mathcal L}$$ (hence the ideals of $${\mathcal L}^{\partial }$$), Y is its set of ideals (the filters of $${\mathcal L}^{\partial }$$) and where $${{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}}^{-1}\;\subseteq Y\times X$$, $$y\;{{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}}^{-1} x$$ iff x$${\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}$$y iff $$x\cap y\neq \emptyset$$. Let $$\phi ^{\prime },\psi ^{\prime }$$ be the generated Galois connection , where for V ⊆ Y, U ⊆ X we have $$\phi ^{\prime }(V)=\{x\in X\;|\; V{{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}}^{-1}x\}$$ = $$\{x\in X\;|\; x{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}} V\}=\psi (V)={ }^{{\mathop{=}^{\kern-5pt\shortmid}}} V$$ and $$\psi ^{\prime }(U)=\{y\in Y\;|\; y\;{{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}}^{-1}U\}=\{y\in Y\;|\; U\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}} \;y\}=\phi (U)= U^{{\mathop{=}^{\kern-5pt\shortmid}}}$$. Therefore, $${\mathcal G}_{\psi ^{\prime }}(X^{\prime })={\mathcal G}_{\phi }(Y)\subseteq (Y)$$ and $${\mathcal G}_{\phi ^{\prime }}(Y^{\prime })={\mathcal G}_{\psi }(X)\subseteq (X)$$. Since $${\mathcal G}_{\psi }(X)\backsimeq{\mathcal G}_{\phi }(Y)^{\partial }$$, we have $${\mathcal G}_{\psi ^{\prime }}(X^{\prime })\backsimeq{\mathcal G}_{\psi }(X)^{\partial }$$. In other words, $$(L^{\partial })_{\sigma }\backsimeq (L_{\sigma })^{\partial }$$. For n-ary maps and product lattices a similar analysis shows that $$({\mathcal L}\times{\mathcal M})_{\sigma }\backsimeq{\mathcal L}_{\sigma }\times{\mathcal M}_{\sigma }$$. Literally, the $$\sigma$$-extension of n-ary maps with mixed monotonicity properties defined in [9] is a map $$f^{\sigma }:{\mathcal G}(Z^{i_{1}})\times \cdots \times{\mathcal G}(Z^{i_{n}})\longrightarrow{\mathcal G}_{\psi }(X)$$, where $${\mathcal G}(Z^{i_{j}})=\left\{\begin{smallmatrix}{\mathcal G}_{\psi }(X) &\ \textrm{when}\ i_{j}=1\\ {\mathcal G}_{\phi }(Y) &\ \textrm{when}\ i_{j}=\partial \end{smallmatrix}\right.$$, which is defined on closed elements by $$f^{\sigma }(\Gamma u_{1},\ldots ,\Gamma u_{n})=\bigwedge \{\Gamma x_{fa_{1}\cdots a_{n}}\;|\;\bigwedge _{j}(a_{j}\in u_{j})\}$$, since $$\alpha (\,fa_{1}\cdots a_{n})=\Gamma x_{fa_{1}\cdots a_{n}}$$. Composing with the dual equivalence $$(\psi ,\phi )$$ at the appropriate argument places we get the equivalent definition of the map $$f^{\sigma }:{\mathcal G}_{\psi }(X)^{i_{1}}\times \cdots \times{\mathcal G}_{\psi }(X)^{i_{n}}\longrightarrow{\mathcal G}_{\psi }(X)$$, where $${\mathcal G}_{\psi }(X)^{i_{j}}=\left\{\begin{smallmatrix}{\mathcal G}_{\psi }(X) & \textrm{when}\ i_{j}=1\\{\mathcal G}_{\psi }(X)^{\partial } & \textrm{when}\ i_{j}=\partial \end{smallmatrix}\right.$$, defined by $$f^{\sigma }(\ldots ,\underbrace{\Gamma u_{j}}_{i_{j}=1},\ldots ,\underbrace{{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}\{u_{r}\}}_{i_{r}=\partial },\ldots )=\bigwedge \{\Gamma x_{fa_{1}\cdots a_{n}}\;|\;\bigwedge _{j}(a_{j}\in u_{j})\}=\Gamma (\bigvee \{x_{fa_{1}\cdots a_{n}}\;|\;\bigwedge _{j}(a_{j}\in u_{j})\})$$. 3 Categories of $$\tau$$-frames 3.1 Relations and operators on base frames Let $$\tau$$ be a similarity type and assume $$\tau _{1},\tau _{\partial }$$ consist of the distribution types in $$\tau$$ of output type 1 and ∂, respectively. We consider frames $$\mathfrak{F}=(X,(R_{\delta })_{\delta \in \tau _{1}},{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}} ,Y,(S^{\partial }_{\delta })_{\delta \in \tau _{\partial }})$$ where (X, $${\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}$$, Y) is a base frame in the sense of Section 2.1 and $$R_{\delta }, S^{\partial }_{\delta }$$ are relations satisfying the following conditions: (R1) If $$\delta =(i_{1},\ldots ,i_{n};i_{n+1})$$, then   \begin{align} \begin{array}{cl} (\textrm{Case}\ \delta\in\tau_{1}) & R_{\delta} \subseteq\; X\times (X^{i_{1}}\times\cdots\times X^{i_{n}})\\ (\textrm{Case}\ \delta\in\tau_{\partial}) & S^{\partial}_{\delta} \subseteq \;Y\times (X^{i_{1}}\times\cdots\times X^{i_{n}}) \end{array} \ \textrm{where}\ X^{i_{j}}\;=\;\left\{ \begin{array}{@{}cl@{}} X & \textrm{if}\ i_{j}=1\\ Y & \textrm{if}\ i_{j}=\partial. \end{array} \right. \end{align} (3) (R2) $$R_{\delta }, S^{\partial }_{\delta }$$ are increasing in the first argument place and decreasing in every other argument place. (R3) For $$\delta \in \tau _{1}$$ and any $$(u_{1},\ldots ,u_{n})\in (X^{i_{1}}\times \cdots \times X^{i_{n}})$$, the set $$R_{\delta } u_{1}\cdots u_{n}$$ is a closed element of $${\mathcal G}_{\psi }(X)$$ (in other words, there exists a point z ∈ X such that for any x ∈ X, $$xR_{\delta } u_{1}\cdots u_{n}$$ iff z ≤ x) and if all $$u_{i}$$ are clopen, then so is $$R_{\delta } u_{1}\cdots u_{n}$$. (R4) For $$\delta \in \tau _{\partial }$$ and any $$(u_{1},\ldots ,u_{n})\in (X^{i_{1}}\times \cdots \times X^{i_{n}})$$, the set $$S^{\partial }_{\delta } u_{1}\cdots u_{n}$$ is a closed element of $${\mathcal G}_{\phi }(Y)$$ (i.e. there is a point v ∈ Y such that for all y ∈ Y, $$yS^{\partial }_{\delta } u_{1}\cdots u_{n}$$ iff v ≤ y) and if all $$u_{i}$$ are clopen, then so is $$S^{\partial }_{\delta } u_{1}\cdots u_{n}$$. Definition 3.1 (Generalized image operators) The relations generate operators, as defined below   \begin{align} &(\textrm{Case}\ \delta\in\tau_{1})\quad\widehat{\bigcirc}{\kern-6.3pt\mid}\ \,_{\delta}(U_{1},\ldots,U_{n})\nonumber\\ &\quad=\{x\in X\;|\; \exists u_{1}\cdots u_{n}\;(xR_{\delta} u_{1}\cdots u_{n}\;\wedge\;\bigwedge_{j}^{i_{j}=1}(u_{j}\in U_{j})\;\wedge\;\bigwedge_{r}^{i_{r}=\partial}(U_{r}\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; u_{r}))\} \end{align} (4)  \begin{align} &(\textrm{Case}\ \delta\in\tau_{\partial})\quad\widehat{\ominus}^{\partial}_{\delta}(V_{1},\ldots,V_{n}) \nonumber\\ &\quad= \{\,y\in Y\;|\; \exists v_{1}\cdots v_{n}\;(yS^{\partial}_{\delta} v_{1}\cdots v_{n}\;\wedge\;\bigwedge_{j}^{i_{j}=1}(v_{j}\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; V_{j})\;\wedge\; \bigwedge_{r}^{i_{r}=\partial}(v_{r}\in V_{r}))\} \end{align} (5)where $$\widehat{\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta }: (X)^{n}\longrightarrow (X)$$ and $$\widehat{\ominus }^{\partial }_{\delta }:(Y)^{n}\longrightarrow (Y)$$. Remark 3.2 The observant reader will notice that $$\widehat{\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta }$$ is the composition of a classical (but two-sorted) image set-operator generated by the relation $$R_{\delta }$$ with the Galois map $$\phi$$ where $$:\prod _{j=1}^{j=n} (Z^{i_{j}})\longrightarrow (X)$$, with $$(Z^{i_{j}})= (X)$$, if $$i_{j}=1$$ and ℘(Y ), when $$i_{j}=\partial$$. A similar observation applies for $$\widehat{\ominus }_{\delta^{\prime}}$$ (with a corresponding two-sorted diamond operator ). Lemma 3.3 Let $${\mathcal G}_{\psi }(X)^{i_{j}}$$ designate $${\mathcal G}_{\psi }(X)$$ if $$i_{j}=1$$, $${\mathcal G}_{\psi }(X)^{\partial }$$ otherwise. Similarly, $${\mathcal G}_{\phi }(Y)^{i_{j}}$$ designates $${\mathcal G}_{\phi }(Y)$$ if $$i_{j}=1$$, $${\mathcal G}_{\phi }(Y)^{\partial }$$ otherwise. If $$\delta \in \tau _{1}$$ and $$U_{j}$$ is a closed element of $${\mathcal G}_{\psi }(X)^{i_{j}}$$, for each j = 1, …, n, then $$\widehat{\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta }(U_{1},\ldots ,U_{n})$$ is a closed element of $${\mathcal G}_{\psi }(X)$$. Similarly, if $$\delta \in \tau _{\partial }$$ and for each r = 1, …, n, $$V_{r}$$ is an open element of $${\mathcal G}_{\phi }(Y)^{i_{r}}$$, then $$\widehat{\ominus }^{\partial }_{\delta }(V_{1},\ldots ,V_{n})\in{\mathcal G}_{\kappa }(Y)$$, hence an open element of $${\mathcal G}_{\phi }(Y)^{\partial }$$. Proof. Recall first (Remark 2.4) that the closed elements of the product $${\mathcal G}_{\psi }(X)^{i_{1}}\times \cdots \times{\mathcal G}_{\psi }(X)^{i_{n}}$$ are tuples of the form $$(\ldots ,\underbrace{\Gamma u_{j}}_{i_{j}=1},\ldots ,\underbrace{{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}\{u_{r}\}}_{i_{r}=\partial },\ldots )$$, with $$u_{j}\in X$$, when $$i_{j}=1$$ and $$u_{r}\in Y$$, when $$i_{r}=\partial$$. For the case $$\delta \in \tau _{1}$$ we verify that $$\widehat{\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta }(\ldots ,\underbrace{\Gamma u_{j}}_{i_{j}=1},\ldots ,\underbrace{{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}\{u_{r}\}}_{i_{r}=\partial },\ldots )=R_{\delta } u_{1}\cdots u_{n}$$. Given definitions, given also that $${ }^{{\mathop{=}^{\kern-5pt\shortmid}}}\{u\}\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; v$$ iff $$v\in \Gamma u$$ iff u ≤ v, the claim reduces to showing that $$xR_{\delta } u_{1}\cdots u_{n}$$ iff $$\exists u^{\prime }_{1}\cdots u^{\prime }_{n}\;(xR_{\delta } u^{\prime }_{1}\cdots u^{\prime }_{n}\;\wedge \;\bigwedge _{i_{j}=1}(u_{j}\leq u^{\prime }_{j})\;\wedge \;\bigwedge _{i_{r}=\partial }(u_{r}\leq u^{\prime }_{r}))$$ iff $$\exists u^{\prime }_{1}\cdots u^{\prime }_{n}\;(xR_{\delta } u^{\prime }_{1}\cdots u^{\prime }_{n}\;\wedge \;\bigwedge _{s}(u_{s}\leq u^{\prime }_{s}))$$. Left to right is obvious and the converse is immediate by the monotonicity properties (R2) of $$R_{\delta }$$. Then $$\widehat{\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta }(\ldots ,\underbrace{\Gamma u_{j}}_{i_{j}=1},\ldots ,\underbrace{{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}\{u_{r}\}}_{i_{r}=\partial },\ldots )=R_{\delta } u_{1}\cdots u_{n}\in{\mathcal G}_{\kappa }(X)$$, by condition (R3) on frames. The argument showing that $$\widehat{\ominus }^{\partial }_{\delta }(\ldots ,\underbrace{\{u_{j}\}{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}}_{i_{j}=1},\ldots ,\underbrace{\Gamma u_{r}}_{i_{r}=\partial },\ldots )=S^{\partial }_{\delta } u_{1}\cdots u_{n}$$ is similar, now using the condition (R4) on frames that $$S^{\partial }_{\delta } u_{1}\cdots u_{n}$$ is a closed element of $${\mathcal G}_{\phi }(Y)$$ (hence an open element of $${\mathcal G}_{\phi }(Y)^{\partial }$$). Corollary 3.4 The generalized image operators $$\widehat{\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta }/\widehat{\ominus }^{\partial }_{\delta }$$ restrict to maps on clopen elements. Proof. Immediate, by Lemma 3.3 and by the restriction in the definition of frames that $$R_{\delta } u_{1}\cdots u_{n}$$ and $$S^{\partial }_{\delta } u_{1}\cdots u_{n}$$ are clopen elements when all the $$u_{i}$$ are clopen. Given the set-operators $$\widehat{\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta }$$ ($$\delta \in \tau _{1}$$) and $$\widehat{\ominus }^{\partial }_{\delta ^{\prime }}$$ ($${\delta ^{\prime }}\in \tau _{\partial }$$), generated by the frame relations $$R_{\delta },S^{\partial }_{\delta ^{\prime }}$$, we now define operators $${\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta },\ominus _{\delta ^{\prime }}$$ on stable sets and their dual operators $${\bigcirc}{\kern-6.4pt\mid}\ \,^{\partial }_{\delta },\ominus ^{\partial }_{\delta ^{\prime }}$$ on co-stable sets, as indicated in the following figure. Definition 3.5 Let $${\bigcirc}{\kern-6.4pt\mid}\ \,^{\partial }_{\delta },\ominus ^{\partial }_{\delta ^{\prime }}:(Y)^{n}\longrightarrow (Y)$$, $$\ominus _{\delta ^{\prime }},{\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta }:(X)^{n}\longrightarrow (X)$$, for $$\delta \in \tau _{1},\;\delta ^{\prime }\in \tau _{\partial }$$, be defined by $${\bigcirc}{\kern-6.4pt\mid}\ \,^{\partial }_{\delta }(V_{1},\ldots ,V_{n}) \;= \;\phi \widehat{\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta }(\psi V_{1},\ldots ,\psi V_{n})$$, for $$V_{j}\subseteq Y$$ $$\ominus _{\delta ^{\prime }}(U_{1},\ldots ,U_{n})=\psi \widehat{\ominus}^{\partial }_{\delta }(\phi U_{1},\ldots ,\phi U_{n})$$, for $$U_{j}\subseteq X$$ $${\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta }=\psi \phi \widehat{\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta }$$ and $$\ominus ^{\partial }_{\delta ^{\prime }}=\phi \psi \widehat{\ominus }_{\delta ^{\prime }}$$. Let also the relations $$R^{\partial }_{\delta }$$ and $$S_{\delta ^{\prime }}$$ be defined be setting   \begin{align} {\hskip7pt}R^{\partial}_{\delta}\subseteq Y\times (X^{i_{1}}\times\cdots\times X^{i_{n}})\ \ \textrm{where}\ \ X^{i_{j}}\;=\;\begin{cases} X & \textrm{if}\ \ i_{j}=1\\ Y & \textrm{if}\ \ i_{j}=\partial \end{cases} \end{align} (6)  \begin{align} {\hskip-85pt}yR^{\partial}_{\delta} u_{1}\cdots u_{n}\equiv&\, \forall x\in X(xR_{\delta} u_{1}\cdots u_{n}\longrightarrow x\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; y)\nonumber\\ \equiv&\, R_{\delta} u_{1}\cdots u_{n}\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; y,\ \textrm{i.e.}\ R^{\partial}_{\delta} u_{1}\cdots u_{n}\;=\;(R_{\delta} u_{1}\cdots u_{n}){{}^{{\mathop{=}^{\kern-5pt\shortmid}}}} \end{align} (7)  \begin{align} {\hskip-10pt}S_{\delta^{\prime}}\subseteq X\times (X^{i_{1}}\times\cdots\times X^{i_{n}})\ \textrm{where}\ X^{i_{j}}=\begin{cases} X & \textrm{if}\ i_{j}=1\\ Y & \textrm{if}\ i_{j}=\partial \end{cases} \end{align} (8)  \begin{align} &xS_{\delta^{\prime}} v_{1}\cdots v_{n}\ \textrm{iff}\quad\ \forall y\in Y\;(yS^{\partial}_{\delta^{\prime}} v_{1}\cdots v_{n}\longrightarrow x\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; y)\ \textrm{iff}\ x\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; S^{\partial}_{\delta^{\prime}} v_{1}\cdots v_{n},\nonumber\\ &\quad\textrm{i.e.}\ S_{\delta^{\prime}} v_{1}\cdots v_{n}\;=\;{}^{{\mathop{=}^{\kern-5pt\shortmid}}}(S^{\partial}_{\delta^{\prime}} v_{1}\cdots v_{n}). \end{align} (9) Lemma 3.6 The following hold, where $$\delta \in \tau _{1},\;\delta ^{\prime }\in \tau _{\partial }$$, $$A_{i}\in{\mathcal G}_{\psi }(X)$$ and $$B_{j}\in{\mathcal G}_{\phi }(Y)$$ 1) $${\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta }(A_{1},\ldots ,A_{n})=\psi \phi (\{x\in X\;|\; \exists u_{1}\cdots u_{n}\;(xR_{\delta } u_{1}\cdots u_{n}\;\wedge \;\bigwedge _{j}^{i_{j}=1}(u_{j}\in A_{j})\;\wedge \;\bigwedge _{r}^{i_{r}=\partial }(A_{r}\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; u_{r}))\})$$$$=\psi \phi \widehat{\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta }(A_{1},\ldots ,A_{n})=\psi {\bigcirc}{\kern-6.4pt\mid}\ \,^{\partial }_{\delta }(\phi A_{1},\ldots ,\phi A_{n})$$ 2) $${\bigcirc}{\kern-6.4pt\mid}\ \,^{\partial }_{\delta }(B_{1},\ldots ,B_{n}) = \{y\in Y\;|\;\forall u_{1}\cdots u_{n}((\bigwedge _{j}^{i_{j}=1}(u_{j}\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; B_{j})\;\wedge \;\bigwedge _{r}^{i_{r}=\partial }(u_{r}\in B_{r}))\;\longrightarrow yR^{\partial }_{\delta } u_{1}\cdots u_{n})\}$$ 3) $$\ominus _{\delta ^{\prime }}(A_{1},\ldots ,A_{n})=\{x\in X\;|\; \forall v_{1}\cdots v_{n}\;((\bigwedge _{j}^{i_{j}=1}(v_{j}\in A_{j})\;\wedge \;\bigwedge _{r}^{i_{r}=\partial }(A_{r}\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; v_{r})\;\longrightarrow \;xS_{\delta ^{\prime }} v_{1}\cdots v_{n}))\}$$ 4) $$\ominus ^{\partial }_{\delta ^{\prime }}(B_{1},\ldots ,B_{n})=\phi \psi (\{y\in Y\;|\; \exists v_{1}\cdots v_{n}\;(yS^{\partial }_{\delta ^{\prime }} v_{1}\cdots v_{n}\;\wedge \;\bigwedge _{j}^{i_{j}=1}(v_{j}\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; V_{j})\;\wedge \; \bigwedge _{r}^{i_{r}=\partial }(v_{r}\in V_{r})\})$$$$=\phi \psi \widehat{\ominus }^{\partial }_{\delta ^{\prime }}(B_{1},\ldots ,B_{n})=\phi \ominus _{\delta ^{\prime }}(\psi B_{1},\ldots ,\psi B_{n})$$ where $$R^{\partial }_{\delta }$$ and $$S_{\delta ^{\prime }}$$ are defined in Definition 3.5. Proof. 1) and 4) are immediate, stated just for explicitness. For 2) we calculate that, for $$B_{i}\in{\mathcal G}_{\phi }(Y)$$, i = 1, …, n,   \begin{align*} &\phi\widehat{\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta}(\psi B_{1},\ldots,\psi B_{n}) \\ &\quad = \{x\in X\;|\; \exists u_{1}\cdots u_{n}\;(xR_{\delta} u_{1}\cdots u_{n}\;\wedge\;\bigwedge_{j}^{i_{j}=1}(u_{j}\in \psi B_{j})\;\wedge\;\bigwedge_{r}^{i_{r}=\partial}(\psi B_{r}\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; u_{r}))\}{^{{\mathop{=}^{\kern-5pt\shortmid}}}}\\ &\quad = \{x\in X\;|\; \exists u_{1}\cdots u_{n}(xR_{\delta} u_{1}\cdots u_{n}\;\wedge\;\bigwedge_{j}^{i_{j}=1}(u_{j}\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; B_{j})\;\wedge\;\bigwedge_{r}^{i_{r}=\partial}(u_{r}\in B_{r}))\}{^{{\mathop{=}^{\kern-5pt\shortmid}}}}\\ &\quad = \{y\in Y\;|\;\forall x\in X((\exists u_{1}\cdots u_{n}(xR_{\delta} u_{1}\cdots u_{n}\;\wedge\bigwedge_{j}^{i_{j}=1}(u_{j}\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; B_{j})\;\wedge\;\bigwedge_{r}^{i_{r}=\partial}(u_{r}\in B_{r}))).\!\!\longrightarrow x\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; y)\}\\ &\quad = \{y\in Y\;|\;\forall x\in X\forall u_{1}\cdots u_{n}(((xR_{\delta} u_{1}\cdots u_{n}\;\wedge\;\bigwedge_{j}^{i_{j}=1}(u_{j}\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; B_{j})\;\wedge\;\bigwedge_{r}^{i_{r}=\partial}(u_{r}\in B_{r}))).\!\!\longrightarrow x\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; y)\}\\ &\quad = \{\,y\in Y\;|\;\forall x\in X\forall u_{1}\cdots u_{n}((\bigwedge_{j}^{i_{j}=1}(u_{j}\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; B_{j})\;\wedge\;\bigwedge_{r}^{i_{r}=\partial}(u_{r}\in B_{r})).\!\longrightarrow(xR_{\delta} u_{1}\cdots u_{n}\longrightarrow x\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; y)\}\\ &\quad = \{\,y\in Y\;|\;\forall u_{1}\cdots u_{n}((\bigwedge_{j}^{i_{j}=1}(u_{j}\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; B_{j})\;\wedge\;\bigwedge_{r}^{i_{r}=\partial}(u_{r}\in B_{r})).\!\longrightarrow\forall x\in X(xR_{\delta} u_{1}\cdots u_{n}\longrightarrow x\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; y)\}\\ &\quad =\{\,y\in Y\;|\;\forall u_{1}\cdots u_{n}((\bigwedge_{j}^{i_{j}=1}(u_{j}\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; B_{j})\;\wedge\;\bigwedge_{r}^{i_{r}=\partial}(u_{r}\in B_{r})).\!\longrightarrow(R_{\delta} u_{1}\cdots u_{n}\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}} \;y)\}\\ &\quad =\{\,y\in Y\;|\;\forall u_{1}\cdots u_{n}((\bigwedge_{j}^{i_{j}=1}(u_{j}\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; B_{j})\;\wedge\;\bigwedge_{r}^{i_{r}=\partial}(u_{r}\in B_{r})).\longrightarrow yR^{\partial}_{\delta} u_{1}\cdots u_{n}\}. \end{align*}2) and 3) are completely similar and we leave details for 3) to the reader. Corollary 3.7 For $$\delta \in \tau _{1}, A_{i}\in{\mathcal G}_{\psi }(X)$$, $${\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta }(A_{1},\ldots ,A_{n})$$ is the join of the $${\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta }(\ldots ,\underbrace{\Gamma u_{j}}_{i_{j}=1},\ldots ,\underbrace{{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}\{u_{r}\}}_{i_{r}=\partial },\ldots )$$ such that for $$i_{j}=1$$, $$u_{j}$$ is in $$A_{j}$$ and for $$i_{r}=\partial$$, $$u_{r}$$ is in $$\phi (A_{r})$$, i.e.   $${\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta}(A_{1},\ldots,A_{n})=\bigvee^{i_{j}=1,u_{j}\in A_{j}}_{i_{r}=\partial,A_{r}\,{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\, u_{r}}R_{\delta} u_{1}\cdots u_{n}= \bigvee^{i_{j}=1,u_{j}\in A_{j}}_{i_{r}=\partial,A_{r}\,{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\, u_{r}}{\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta}(\ldots,\underbrace{\Gamma u_{j}}_{i_{j}=1},\ldots,\underbrace{{}^{{\mathop{=}^{\kern-5pt\shortmid}}}\{u_{r}\}}_{i_{r}=\partial},\ldots).$$Similarly, for $$\delta ^{\prime }\in \tau _{\partial }, B_{i}\in{\mathcal G}_{\phi }(Y)$$  $$\ominus^{\partial}_{\delta^{\prime}}(B_{1},\ldots,B_{n})=\bigvee^{i_{j}=1,v_{j}\,{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\, B_{j}}_{i_{r}=\partial,v_{r}\in B_{r}}S^{\partial}_{\delta^{\prime}}v_{1}\cdots v_{n}=\bigvee^{i_{j}=1,v_{j}\,{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\, B_{j}}_{i_{r}=\partial,v_{r}\in B_{r}}\ominus^{\partial}_{\delta^{\prime}}(\ldots,\underbrace{\{v_{j}\}{}^{{\mathop{=}^{\kern-5pt\shortmid}}}}_{i_{j}=1},\ldots,\underbrace{\Gamma v_{r}}_{i_{r}=\partial},\ldots).$$ Proof.   \begin{align*} {\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta}(A_{1},\ldots,A_{n})&=\psi\phi\bigcup\nolimits_{i_{r}=\partial,A_{r}\,{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\, u_{r}}^{i_{j}=1,u_{j}\in A_{j}}R_{\delta} u_{1}\cdots u_{n} &\textrm{by definition}\\ & =\bigvee\nolimits^{i_{j}=1,u_{j}\in A_{j}}_{i_{r}=\partial,A_{r}\,{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\, u_{r}}\psi\phi R_{\delta} u_{1}\cdots u_{n}&\\ &=\bigvee\nolimits_{i_{r}=\partial,A_{r}\,{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\, u_{r}}^{i_{j}=1,u_{j}\in A_{j}}R_{\delta} u_{1}\cdots u_{n}&R_{\delta} u_{1}\cdots u_{n}\ \textrm{is stable}\\ &=\bigvee\nolimits_{i_{r}=\partial,A_{r}\,{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\, u_{r}}^{i_{j}=1,u_{j}\in A_{j}}\widehat{\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta}(\ldots,\underbrace{\Gamma u_{j}}_{i_{j}=1},\ldots,\underbrace{{}^{{\mathop{=}^{\kern-5pt\shortmid}}}\{u_{r}\}}_{i_{r}=\partial},\ldots)&\textrm{Lemma 3.3}\\ &=\bigvee\nolimits_{i_{r}=\partial,A_{r}\,{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\, u_{r}}^{i_{j}=1,u_{j}\in A_{j}}{\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta}(\ldots,\underbrace{\Gamma u_{j}}_{i_{j}=1},\ldots,\underbrace{{}^{{\mathop{=}^{\kern-5pt\shortmid}}}\{u_{r}\}}_{i_{r}=\partial},\ldots)\!&\!\!\!\textrm{definition of}\ \, {\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta}, \textrm{Lemma 3.3}. \end{align*}The proof for $$\ominus ^{\partial }_{\delta ^{\prime }}$$ is completely similar, left to the reader. The following is then an immediate consequence, tightening the result of Corollary 2. Corollary 3.8 The operators $${\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta },{\bigcirc}{\kern-6.4pt\mid}\ \,^{\partial }_{\delta }$$ and $$\ominus _{\delta ^{\prime }},\ominus ^{\partial }_{\delta ^{\prime }}$$, where $$\delta \in \tau _{1}, \delta ^{\prime }\in \tau _{\partial }$$ restrict to maps on clopen elements of $${\mathcal G}_{\psi }(X)$$ and of $${\mathcal G}_{\phi }(Y)$$, accordingly. As the reader can easily verify, each of $${\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta },\ominus _{\delta ^{\prime }}$$ has the monotonicity properties corresponding to the distribution types $$\delta ,\delta ^{\prime }$$. However, additional axioms must be assumed to ensure that the operators are normal, of distribution type $$\delta ,\delta ^{\prime }$$, respectively. We add appropriate axioms in our definition of $$\tau$$-frames, in Section 3.2. 3.2 $$\tau$$-Frames Let $$\tau =(\tau _{1},\tau _{\partial })$$ be a similarity type with distribution types $$\delta \in \tau _{1}$$ of output type 1 and $$\delta ^{\prime }$$ in $$\tau _{\partial }$$ of output type ∂. Definition 3.9 ($$\tau$$-Frames) A Kripke–Galois 2-sorted $$\tau$$-frame $$\mathfrak{F}=(X,(R_{\delta },R^{\partial }_{\delta })_{\delta \in \tau _{1}},{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}} ,Y,(S_{\delta },S^{\partial }_{\delta })_{\delta \in \tau _{\partial }})$$ is a structure where (X, $${\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}$$, Y) satisfies axioms (F1–F5) for lattice frames, presented in Section 2.2, and the relations $$R_{\delta }$$ with $$\delta \in \tau _{1}$$, $$S^{\partial }_{\delta }$$ with $$\delta \in \tau _{\partial }$$ satisfy axioms (R1–R4), presented in Section 3.1, as well as axioms R5–R8 stated below: (R5) $$R^{\partial }_{\delta }\subseteq Y\times (X^{i_{1}}\times \cdots \times X^{i_{n}})\ \textrm{with}\ X^{i_{j}}\;=\;\left\{\begin{smallmatrix} X & \textrm{if}\ i_{j}=1\\ Y & \textrm{if}\ i_{j}=\partial \end{smallmatrix}\right.$$ and $$\\\forall (u_{1},\ldots , u_{n})\in (X^{i_{1}}\times \cdots \times X^{i_{n}})\;R^{\partial }_{\delta } u_{1}\cdots u_{n}\;=\;(R_{\delta } u_{1}\cdots u_{n}){ }^{{\mathop{=}^{\kern-5pt\shortmid}}}$$ (R6) $$S_{\delta ^{\prime }}\subseteq X\times (X^{i_{1}}\times \cdots \times X^{i_{n}})\ \textrm{with}\ X^{i_{j}}=\left\{\begin{smallmatrix} X & \textrm{if}\ i_{j}=1\\ Y & \textrm{if}\ i_{j}=\partial \end{smallmatrix}\right.$$ and $$\\\forall (v_{1},\ldots ,v_{n})\in (X^{i_{1}}\times \cdots \times X^{i_{n}})\;S_{\delta ^{\prime }} v_{1}\cdots v_{n}\;=\;{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}(S^{\partial }_{\delta ^{\prime }} v_{1}\cdots v_{n})$$ (R7) For all x ∈ X and all $$u_{s},u^{\prime }_{s}\in X^{i_{s}}$$, for all j, s = 1, …, n,   $$xR_{\delta} u_{1}\cdots (u_{j}\cap u^{\prime}_{j})\cdots u_{n}\;\longrightarrow\;\forall y\in Y\;(yR^{\partial}_{\delta} u_{1}\cdots u_{j}\cdots u_{n}\;\wedge\;yR^{\partial}_{\delta} u_{1}\cdots u^{\prime}_{j}\cdots u_{n}\;\longrightarrow\;x\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; y)$$ (R8) For all y ∈ Y and all $$u_{s},u^{\prime }_{s}\in X^{i_{s}}$$, for r, s = 1, …, n,   $$yS^{\partial}_{\delta} u_{1}\cdots (u_{r}\cap u^{\prime}_{r})\cdots u_{n}\;\longrightarrow\;\forall x\in X\;(yS_{\delta} u_{1}\cdots u_{r}\cdots u_{n}\;\wedge\;yS_{\delta} u_{1}\cdots u^{\prime}_{r}\cdots u_{n}\;\longrightarrow\;x\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; y)$$ A Kripke–Galois 2-sorted $$\tau$$-frame will be referred to in the sequel as simply a $$\tau$$-frame. $$\tau$$-frame morphisms are the morphisms $$(\,f,h):(X_{1},{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}} _{1},Y_{1})\longrightarrow (X_{2},{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}} _{2},Y_{2})$$ of their underlying lattice frames (Definition 2.6) where, in addition, $$f^{\ast }=f^{-1},h^{\ast }=h^{-1}$$ are homomorphisms of the respective $$\tau$$-algebras of clopens. We have opted to define $$\tau$$-frames with a pair of relations $$(R_{\delta }, R^{\partial }_{\delta })$$, $$(S_{\delta ^{\prime }},S^{\partial }_{\delta ^{\prime }})$$, for $$\delta \in \tau _{1},\delta ^{\prime }\in \tau _{\partial }$$ so as to be consistent with our definition of Kripke–Galois frames in [14, 16], hence we added axioms (R5, R6) to $$\tau$$-frames. Axioms (R7–R8) are included to ensure that the operators generated by the relations are normal in the sense of the next Proposition. Proposition 3.10 The operator $${\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta }:\prod _{j=1}^{j=n}{\mathcal G}_{\kappa }({\mathcal G}_{\psi }(X)^{i_{j}})\longrightarrow{\mathcal G}_{\kappa }(X)$$ is a normal operator of distribution type $$\delta =(i_{1},\ldots ,i_{n};1)\in \tau _{1}$$, where $${\mathcal G}_{\psi }(X)^{i_{j}}={\mathcal G}_{\psi }(X)$$ when $$i_{j}=1$$ and $${\mathcal G}_{\psi }(X)^{\partial }$$ when $$i_{j}=\partial$$ and $${\mathcal G}_{\kappa }({\mathcal G}_{\psi }(X)^{i_{j}})={\mathcal G}_{\kappa }(X)$$ or it is $${\mathcal G}_{o}(X)$$, respectively. Similarly for $$\ominus _{\delta ^{\prime }}, \delta ^{\prime }\in \tau _{\partial }$$. Proof. We separate cases. Case $$i_{j}=1=i_{n+1}$$: We need to show distribution of $$\ {\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta }$$ over joins at the j-th position, where $$X^{i_{j}}=X$$ and then $$u_{j},u^{\prime }_{j}\in X$$. It suffices to establish that   $${\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta}(\ldots,\Gamma u_{j}\vee\Gamma u^{\prime}_{j},\ldots)\subseteq{\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta}(\ldots,\Gamma u_{j},\ldots)\vee{\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta}(\ldots,\Gamma u^{\prime}_{j},\ldots)$$since the other direction follows from monotonicity. Closed elements form a lattice and this induces a lattice structure on each of X, Y, where $$\Gamma u_{j}\vee \Gamma u^{\prime }_{j}=\Gamma (u_{j}\cap u^{\prime }_{j})$$. Hence we need to show that   $${\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta}(\ldots,\Gamma(u_{j}\cap u^{\prime}_{j}),\ldots)\subseteq{\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta}(\ldots,\Gamma u_{j},\ldots)\vee{\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta}(\ldots,\Gamma u^{\prime}_{j},\ldots).$$By Lemma 3.3, definition of $$\,{\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta }$$ and by the frame axiom (R3), the above requirement is equivalent to the following:   \begin{align*} R_{\delta} u_{1}\cdots (u_{j}\cap u^{\prime}_{j})\cdots u_{n} &\subseteq R_{\delta} u_{1}\cdots u_{j}\cdots u_{n}\;\vee\;R_{\delta} u_{1}\cdots u^{\prime}_{j}\cdots u_{n} \\ &=\psi\phi(R_{\delta} u_{1}\cdots u_{j}\cdots u_{n}\;\cup\;R_{\delta} u_{1}\cdots u^{\prime}_{j}\cdots u_{n})\\ &=\psi((R_{\delta} u_{1}\cdots u_{j}\cdots u_{n}){}^{{\mathop{=}^{\kern-5pt\shortmid}}}\cap (R_{\delta} u_{1}\cdots u^{\prime}_{j}\cdots u_{n}){^{{\mathop{=}^{\kern-5pt\shortmid}}}})\\ &=\psi(R^{\partial}_{\delta} u_{1}\cdots u_{j}\cdots u_{n}\;\cap\;R^{\partial}_{\delta} u_{1}\cdots u^{\prime}_{j}\cdots u_{n}) \end{align*}where we used axiom (R5). Hence the desired inclusion is equivalent to   \begin{align*} xR_{\delta} u_{1}\cdots (u_{j}\cap u^{\prime}_{j})\cdots u_{n}&\longrightarrow x\; {\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\;(R^{\partial}_{\delta} u_{1}\cdots u_{j}\cdots u_{n}\;\cap\;R^{\partial}_{\delta} u_{1}\cdots u^{\prime}_{j}\cdots u_{n}), \text{i.e.}\\ xR_{\delta} u_{1}\cdots (u_{j}\cap u^{\prime}_{j})\cdots u_{n}&\longrightarrow \forall y\in Y\;(yR^{\partial}_{\delta} u_{1}\cdots u_{j}\cdots u_{n}\;\wedge\;yR^{\partial}_{\delta} u_{1}\cdots u^{\prime}_{j}\cdots u_{n}\;\longrightarrow\;x\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; y) \end{align*}which is precisely axiom (R7). Case$$i_{r}=\partial \neq 1= i_{n+1}$$: We need to show co-distribution of $${\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta }$$ over meets at the r-th position, where $$X^{i_{r}}=Y$$, in other words the following needs to be proven, where $$u_{r},u^{\prime }_{r}\in Y$$.   $${\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta}\left(\ldots,{}^{{\mathop{=}^{\kern-5pt\shortmid}}}\{u_{r}\}\cap{}^{{\mathop{=}^{\kern-5pt\shortmid}}}\{u^{\prime}_{r}\},\ldots\right)={\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta}\left(\ldots,{}^{{\mathop{=}^{\kern-5pt\shortmid}}}\{u_{r}\},\ldots\right)\vee{\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta}\left(\ldots,{}^{{\mathop{=}^{\kern-5pt\shortmid}}} \{u^{\prime}_{r}\},\ldots\right).$$Note that $$\phi ({ }^{{\mathop{=}^{\kern-5pt\shortmid}}}\{u_{r}\}\cap{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}\{u^{\prime }_{r}\})=\phi ({ }^{{\mathop{=}^{\kern-5pt\shortmid}}}\{u_{r}\})\vee \phi ({ }^{{\mathop{=}^{\kern-5pt\shortmid}}}\{u^{\prime }_{r}\})=\Gamma u_{r}\vee \Gamma u^{\prime }_{r}=\Gamma (u_{r}\cap u^{\prime }_{r})$$ and therefore $${ }^{{\mathop{=}^{\kern-5pt\shortmid}}}\{u_{r}\}\cap{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}\{u^{\prime }_{r}\}=\psi (\Gamma (u_{r}\cap u^{\prime }_{r}))={ }^{{\mathop{=}^{\kern-5pt\shortmid}}}\{u_{r}\cap u^{\prime }_{r}\}$$. Given Lemma 3.3, given the definition of $$\,{\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta }$$, given also the frame axiom (R3) and the monotonicity properties of the relation $$R_{\delta }$$ prescribed in the frame axiom (R2), it suffices to show that   $$R_{\delta} u_{1}\cdots\left(u_{r}\cap u^{\prime}_{r}\right)\cdots u_{n}\;\subseteq\; R_{\delta} u_{1}\cdots u_{r}\cdots u_{n}\;\vee\;R_{\delta} u_{1}\cdots u^{\prime}_{r}\cdots u_{n}.$$We have $$R_{\delta } u_{1}\cdots u_{r}\cdots u_{n}\;\vee \;R_{\delta } u_{1}\cdots u^{\prime }_{r}\cdots u_{n}=\psi (R^{\partial }_{\delta } u_{1}\cdots u_{r}\cdots u_{n}\;\cap \;R^{\partial }_{\delta } u_{1}\cdots u^{\prime }_{r}\cdots u_{n})$$, by the same argument as in Case 1 above. The rest follows by the frame axiom (R7). Case$$i_{r}=\partial =i_{n+1}$$ Axiom (R8) implies, by an analogous argument to the case $$i_{j}=1=i_{n+1}$$, that the dual operator $$\ominus ^{\partial }_{\delta }$$ distributes over joins in $${\mathcal G}_{\kappa }(Y)$$ at the r-th position. Indeed, axiom (R8) replaces $$R_{\delta }$$ with $$S^{\partial }_{\delta }$$ and interchanges the roles of X, Y in the statement of the condition. Then use duality of the operators $$\ominus _{\delta },\ominus ^{\partial }_{\delta }$$ to derive the desired conclusion. Case$$i_{r}=1\neq \partial =i_{n+1}$$ As in the previous case, axiom (R8) implies, by an argument analogous to the case $$i_{j}=\partial \neq 1=i_{n+1}$$, that $$\ominus ^{\partial }_{\delta }$$ co-distributes over intersections, turning them to joins in $${\mathcal G}_{\kappa }(Y)$$. Then use again the duality of the operators $$\ominus _{\delta },\ominus ^{\partial }_{\delta }$$ to derive the desired conclusion. By Corollary 3.8, the lattice of clopen elements is closed under the operators. Hence we conclude: Corollary 3.11 Each of $$\,{\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta },\ominus _{\delta ^{\prime }}$$ is a normal operator of $${\mathcal G}_{\kappa o}(X)$$ and similarly for the dual operators $${\bigcirc}{\kern-6.4pt\mid}\ \,^{\partial }_{\delta },\ominus ^{\partial }_{\delta ^{\prime }}$$ and $${\mathcal G}_{\kappa o}(Y)$$. The class of $$\tau$$-frames specified above is too large to allow for proving duality with normal lattice expansions. For duality purposes, we distinguish a subclass that satisfies, in addition, axioms (R9, R10) stated below. (R9) For each $$\delta \in \tau _{1}$$ and all $$(u_{1},\ldots ,u_{n})\in X^{i_{1}}\times \cdots \times X^{i_{n}}$$ and z ∈ X, if for all clopen elements $$u^{s}_{*}$$ below $$u_{s}$$, for each s = 1, …, n, there exist $$(v_{1},\ldots ,v_{n})\in X^{i_{1}}\times \cdots \times X^{i_{n}}$$ such that $$zR_{\delta } v_{1}\cdots v_{n}$$ and $$u^{s}_{*}\leq v_{s}$$, for each s, then there exist $$(v^{\prime }_{1},\ldots ,v^{\prime }_{n})\in X^{i_{1}}\times \cdots \times X^{i_{n}}$$ such that $$zR_{\delta } v^{\prime }_{1}\cdots v^{\prime }_{n}$$ and $$u_{s}\leq v^{\prime }_{s}$$, for each s. (R10) For each $$\delta ^{\prime }\in \tau _{\partial }$$ and all $$(u_{1},\ldots ,u_{n})\in X^{i_{1}}\times \cdots \times X^{i_{n}}$$ and y ∈ Y, if for all clopen elements $$u^{s}_{*}$$ below $$u_{s}$$, for each s = 1, …, n, there exist $$(v_{1},\ldots ,v_{n})\in X^{i_{1}}\times \cdots \times X^{i_{n}}$$ such that $$yS^{\partial }_{\delta ^{\prime }} v_{1}\cdots v_{n}$$ and $$u^{s}_{*}\leq v_{s}$$, for each s, then there exist $$(v^{\prime }_{1},\ldots ,v^{\prime }_{n})\in X^{i_{1}}\times \cdots \times X^{i_{n}}$$ such that $$yS^{\partial }_{\delta ^{\prime }} v^{\prime }_{1}\cdots v^{\prime }_{n}$$ and $$u_{s}\leq v^{\prime }_{s}$$, for each s. Remark 3.12 ($$\tau$$-Frames) Henceforth, by a $$\tau$$-frame we mean a frame also satisfying axioms (R9, R10). Proposition 3.13 In a $$\tau$$-frame (satisfying, in addition, axioms (R9, R10), by the previous Remark) the operators $${\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta },\ominus _{\delta ^{\prime }}$$ are completely normal in $${\mathcal G}_{\psi }(X)$$, with distribution types $$\delta$$ and $$\delta ^{\prime }$$, respectively. Their duals $${\bigcirc}{\kern-6.4pt\mid}\ \,^{\partial }_{\delta },\ominus ^{\partial }_{\delta ^{\prime }}$$ are completely normal operators in $${\mathcal G}_{\phi }(Y)$$ with distribution types the duals $$\overline{\delta }, \overline{\delta }^{\prime }$$, respectively. Proof. Recall from Lemma 2.5 that $$(\imath ,{\mathcal G}_{\psi }(X))$$ is a canonical extension of $${\mathcal G}_{\kappa o}(X)$$. Given Corollary 3.7 and the fact that $$\sigma$$-extensions on stable sets are defined using join-density of closed elements, the claim of complete normality of the operators follows if we can show that their restriction to closed elements is the $$\sigma$$-extension of their restriction to clopens. Axioms (R9, R10) are assumed in order to enforce that. The claim to be verified for the case $$\delta \in \tau _{1}$$ is the following, assuming for simplicity of the argument that we designate clopen elements by placing a low index *, as in $$x^{\,j}_{*}$$, $$y^{\,r}_{*}$$,   \begin{align*} &{\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta}(\ldots,\underbrace{\Gamma u_{j}}_{i_{j}=1},\ldots,\underbrace{{}^{{\mathop{=}^{\kern-5pt\shortmid}}}\{u_{r}\}}_{i_{r}=\partial},\ldots)\\ &\quad= \bigcap_{i_{j}=1,u^{j}_{*}\in{\mathcal G}_{\kappa o}(X)}^{i_{r}=\partial,u^{r}_{*}\in{\mathcal G}_{\kappa o}(Y)}\{{\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta}(\ldots,\underbrace{\Gamma u_{*}^{\,j}}_{i_{j}=1},\ldots,\underbrace{{}^{{\mathop{=}^{\kern-5pt\shortmid}}}\{u_{*}^{r}\}}_{i_{r}=\partial},\ldots) \;\Big|\; \bigwedge_{i_{j}=1}(\Gamma u_{j}\subseteq\Gamma u_{*}^{\,j})\wedge\bigwedge_{i_{r}=\partial}({}^{{\mathop{=}^{\kern-5pt\shortmid}}}\{u_{*}^{r}\}\subseteq{}^{{\mathop{=}^{\kern-5pt\shortmid}}}\{u_{r}\}) \} \end{align*}since the expression to the right of equality defines precisely the $$\sigma$$-extension $${\bigcirc}{\kern-6.4pt\mid}\ \,^{\sigma }_{\delta }$$ of the restriction of the operator $${\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta }$$ to clopens. The inclusion left-to-right follows from monotonicity properties, hence the desired inclusion, slightly restated, is the following:   $$\bigcap_{i_{j}=1,u^{j}_{*}\leq u_{j}, u^{\,j}_{*}\in{\mathcal G}_{\kappa o}(X)}^{i_{r}=\partial,u^{r}_{*}\leq u_{r},u^{r}_{*}\in{\mathcal G}_{\kappa o}(Y)}{\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta}(\ldots,\underbrace{\Gamma u_{*}^{j}}_{i_{j}=1},\ldots,\underbrace{{}^{{\mathop{=}^{\kern-5pt\shortmid}}}\{u_{*}^{r}\}}_{i_{r}=\partial},\ldots) \;\;\subseteq \;\;{\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta}(\ldots,\underbrace{\Gamma u_{j}}_{i_{j}=1},\ldots,\underbrace{{}^{{\mathop{=}^{\kern-5pt\shortmid}}}\{u_{*}^{r}\}}_{i_{r}=\partial},\ldots)$$which is equivalent to the claim that for all z ∈ X, if for all clopen elements $$u^{s}_{*}$$ below $$u_{s}$$, for each s = 1, …, n, there exist $$(v_{1},\ldots ,v_{n})\in X^{i_{1}}\times \cdots \times X^{i_{n}}$$ such that $$zR_{\delta } v_{1}\cdots v_{n}$$ and $$u^{s}_{*}\leq v_{s}$$, for each s, then there exist $$(v^{\prime }_{1},\ldots ,v^{\prime }_{n})\in X^{i_{1}}\times \cdots \times X^{i_{n}}$$ such that $$zR_{\delta } v^{\prime }_{1}\cdots v^{\prime }_{n}$$ and $$u_{s}\leq v^{\prime }_{s}$$ for each s = 1, …, n. But this is precisely axiom (R9). For the case $$\delta ^{\prime }\in \tau _{\partial }$$, we merely prove the claim for the dual operator $$\ominus ^{\partial }_{\delta ^{\prime }}$$, by dualising the argument and using axiom (R10). The corresponding claim for $$\ominus _{\delta ^{\prime }}$$ follows from the fact that $$\ominus _{\delta ^{\prime }}$$ is the dual of $$\ominus ^{\partial }_{\delta ^{\prime }}$$. To be precise, the argument outlined above shows that $$\ominus _{\delta ^{\prime }}$$ is the $$\pi$$-extension of its restriction to clopens. However, by normality (Proposition 1) combining with the results of [9] (Lemmas 4.3, 4.4, 4.6) $$\sigma$$ and $$\pi$$-extensions coincide for normal operators. 4 Stone duality for normal lattice expansions In this section we show that the duality of the categories of bounded lattices and ⊥-frames proven in [19] lifts to a duality of the categories of normal lattice expansions of similarity type $$\tau$$ with the corresponding categories of $$\tau$$-frames. Let $$\mathfrak{F}_{\tau }$$ be a $$\tau$$-frame. Its complex algebra $$\mathfrak{F}^{+}_{\tau }=\hat{C}(\mathfrak{F}_{\tau })$$ is the algebra of clopens of $${\mathcal G}_{\psi }(X)$$ (which is a subalgebra of the full complex algebra of stable sets of the frame) and its dual complex algebra is the algebra of clopens of $${\mathcal G}_{\phi }(Y)$$. By Corollary 3.13, the lattice of clopens with the operators $${\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta },\ominus _{\delta ^{\prime }}$$ generated by the relations $$R_{\delta },S_{\delta ^{\prime }}$$, with $$\delta \in \tau _{1},\delta ^{\prime }\in \tau _{\partial }$$, is a normal lattice expansion, hence an object of $$\textbf{L}_{\tau }$$. Conversely, we consider lattice expansions $${\mathcal A}_{\tau }=({\mathcal L},(f_{\delta })_{\delta \in \tau _{1}},(h_{\delta })_{\delta \in \tau _{\partial }})$$ with normal operators $$(f_{\delta })_{\delta \in \tau _{1}}$$ and $$(h_{\delta })_{\delta \in \tau _{\partial }}$$, where $${\mathcal L}=(L,\wedge ,\vee ,0,1)$$ is the underlying bounded lattice and we construct their dual frame $$\hat{F}({\mathcal A}_{\tau })=({\mathcal A}_{\tau })_{+}$$. Let (X, $${\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}$$, Y) be the dual frame of the lattice, after [19] (hence satisfying the frame axioms (F1–F5)), where X, Y are the sets of filters and ideals, respectively of the lattice and $${\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}$$ ⊆ X × Y is defined by x$${\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}$$y iff $$\,x\cap y\neq \emptyset$$. To extend the lattice frame to a $$\tau$$-frame we rely on the definitions of filter/ideal operators we introduced in [12]. Case$$\,\delta \in \tau _{1}$$: For each $$\delta =(i_{1},\ldots ,i_{n};1)\in \tau _{1}$$ and normal lattice operator $$f_{\delta }:{\mathcal L}^{i_{1}}\times \cdots \times{\mathcal L}^{i_{n}}\longrightarrow{\mathcal L}$$ define the relation $$R_{\delta }\subseteq X\times (X^{i_{1}}\times \cdots \times X^{i_{n}})\ \textrm{where}\ X^{i_{j}}=\left\{\begin{smallmatrix}\!\! X & \textrm{if}\ i_{j}=1\\ \!\! Y & \textrm{if}\ i_{j}=\partial \end{smallmatrix}\right.$$ by setting $$xR_{\delta } u_{1}\cdots u_{n}\;\ \textrm{iff}\ \;f^{\flat }_{\delta }(u_{1},\ldots ,u_{n})\leq x$$, where ≤ designates filter inclusion and $$f^{\flat }_{\delta}\!:X^{i_{1}}\times \cdots \times X^{i_{n}}\!\longrightarrow\! X$$, where $$X^{i_{j}}=X$$ if $$i_{j}=1$$ and $$X^{i_{j}}=Y$$ when $$i_{j}=\partial$$, is defined by (10), after [12],   \begin{align} f^{\flat}_{\delta}(u_{1},\ldots,u_{n})=\bigvee\left\{x_{fa_{1}\cdots a_{n}}\;\Big|\; \bigwedge_{j}(a_{j}\in u_{j})\right\} \end{align} (10)where we systematically designate principal filters by $$x_{e}=e\uparrow$$ and principal ideals by $$y_{e}=e\downarrow$$. The relation $$R^{\partial }_{\delta }\subseteq Y\times (X^{i_{1}}\times \cdots \times X^{i_{n}})$$ is defined by condition (7), now taking the form   $$yR^{\partial}_{\delta} u_{1}\cdots u_{n}\ \textrm{iff}\ R_{\delta} u_{1}\cdots u_{n}\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; y\ \textrm{iff}\ f^{\flat}_{\delta}(u_{1},\ldots,u_{n})\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; y.$$ Case$$\,\delta ^{\prime }\in \tau _{\partial }$$: Similarly, for each $$\delta ^{\prime }\in \tau _{\partial }$$ and normal lattice operator $$h_{\delta ^{\prime }}:{\mathcal L}^{i_{1}}\times \cdots \times{\mathcal L}^{i_{n}}\longrightarrow{\mathcal L}^{\partial }$$, define $$S^{\partial }_{\delta ^{\prime }}\subseteq Y\times (X^{i_{1}}\times \cdots \times X^{i_{n}})$$, where $$X^{i_{s}}=X$$ if $$i_{s}=1$$ and $$X^{i_{s}}=Y$$ if $$i_{s}=\partial$$, by setting $$yS^{\partial }_{\delta ^{\prime }}u_{1}\cdots u_{n}$$ iff $$h^{\sharp }_{\delta ^{\prime }} (u_{1},\ldots ,u_{n})\leq y$$, where $$h^{\sharp }_{\delta ^{\prime }}:X^{i_{1}}\times \cdots \times X^{i_{n}}\longrightarrow Y$$, where $$X^{i_{s}}=X$$, when $$i_{s}=1$$ and $$X^{i_{s}}=Y$$ when $$i_{s}=\partial$$ is defined by equation (11), after [12],   \begin{align} h^{\sharp}_{\delta^{\prime}}(u_{1},\ldots,u_{n})=\bigvee\left\{y_{ha_{1}\cdots a_{n}}\;\Big|\; \bigwedge_{j}(a_{j}\in u_{j})\right\} \end{align} (11)where recall that $$y_{e}$$ designates a principal ideal. The definition of the relation $$S_{\delta ^{\prime }}$$ in (9) now takes the form   $$xS_{\delta^{\prime}}u_{1}\cdots u_{n}\ \textrm{iff}\ x\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; S^{\partial}_{\delta^{\prime}}u_{1}\cdots u_{n} \textrm{ iff}\ x\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; h^{\sharp}_{\delta^{\prime}}(u_{1},\ldots,u_{n}).$$ By definition of the canonical relations $$R_{\delta }, S^{\partial }_{\delta ^{\prime }}$$ the frame axiom (R1) holds for the canonical frame. Theorem 4.1 (1) The canonical relations $$R_{\delta }, S^{\partial }_{\delta ^{\prime }}$$ are increasing in the first place and decreasing in every other place (frame axiom (R2)). (2) For any $$(u_{1},\ldots ,u_{n})\in X^{i_{1}}\times \cdots \times X^{i_{n}}$$, $$R_{\delta } u_{1}\cdots u_{n}$$ is a closed element of $${\mathcal G}_{\psi }(X)$$ and $$S^{\partial }_{\delta ^{\prime }}u_{1}\cdots u_{n}$$ is a closed element of $${\mathcal G}_{\phi }(Y)$$. Furthermore, if all the $$u_{i}$$ are clopen points, then so is each of $$R_{\delta } u_{1}\cdots u_{n}, S^{\partial }_{\delta ^{\prime }}u_{1}\cdots u_{n}$$ (frame axioms (R3, R4)). (3) For each $$\delta \in \tau _{1}, \delta ^{\prime }\in \tau _{\partial }$$, $${\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta }=f_{\delta }^{\sigma }$$ and $$\ominus _{\delta ^{\prime }}=h^{\sigma }_{\delta ^{\prime }}$$. (4) The canonical frame is a $$\tau$$-frame, i.e. axioms (F1–F5) and (R1–R10) hold. Proof. For 1), each of $$f^{\flat }_{\delta }, h^{\sharp }_{\delta ^{\prime }}$$ is monotone in each argument place. From this fact and from the definition the monotonicity properties claimed for the canonical relations follow. For 2), it is immediate from the definition of the relations that   \begin{align} R_{\delta} u_{1}\cdots u_{n}=\{x\in X\;|\; xR_{\delta} u_{1}\cdots u_{n}\}=\{x\in X\;|\; f^{\flat}_{\delta}(u_{1},\ldots,u_{n})\leq x\}=\Gamma(\,f^{\flat}_{\delta}(u_{1},\ldots,u_{n})) \end{align} (12)hence $$R_{\delta } u_{1}\cdots u_{n}$$ is a closed element of $${\mathcal G}_{\psi }(X)$$. By [12], Lemma 6.7, $$f^{\flat }_{\delta }$$ preserves principal filters/ideals, hence $$R_{\delta }$$ is clopen, when all the $$u_{i}$$ are clopen points (principal filters, or principal ideals). Similarly,   \begin{align} S^{\partial}_{\delta^{\prime}}u_{1}\cdots u_{n}=\Gamma(h^{\sharp}_{\delta^{\prime}}(u_{1},\ldots,u_{n}))\qquad xS_{\delta^{\prime}}u_{1}\cdots u_{n}\equiv x\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; h^{\sharp}_{\delta^{\prime}}(u_{1},\ldots,u_{n}) \end{align} (13)hence $$S^{\partial }_{\delta ^{\prime }}$$ is a closed element of $${\mathcal G}_{\phi }(Y)$$ and, by [12], Lemma 6.7, $$h^{\sharp }_{\delta ^{\prime }}$$ preserves principal filters/ideals, hence $$S^{\partial }_{\delta ^{\prime }}$$ is clopen, when all the $$u_{i}$$ are clopen points. For 3), by the proof of Lemma 3.3 (instantiated to the canonical frame, already verified to satisfy axioms (F1–F5) and (R1–R4)) and by equation (12), $$\widehat{\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta }(\ldots ,\underbrace{\Gamma u_{j}}_{i_{j}=1},\ldots ,\underbrace{{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}\{u_{r}\}}_{i_{r}=\partial },\ldots )=R_{\delta } u_{1}\cdots u_{n}=\Gamma (\,f^{\flat }_{\delta }(u_{1},\ldots ,u_{n}))$$, hence we obtain $$\psi \phi \widehat{\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta }(\ldots ,\underbrace{\Gamma u_{j}}_{i_{j}=1},\ldots ,\underbrace{{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}\{u_{r}\}}_{i_{r}=\partial },\ldots )={\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta }(\ldots ,\underbrace{\Gamma u_{j}}_{i_{j}=1},\ldots ,\underbrace{{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}\{u_{r}\}}_{i_{r}=\partial },\ldots )=\Gamma (\,f^{\flat }_{\delta }(u_{1},\ldots ,u_{n}))$$. To complete the proof of claim 3), the following lemma is needed. Lemma 4.2 $$\Gamma (f^{\flat }_{\delta }(u_{1},\ldots ,u_{n}))=f^{\sigma }_{\delta }(\ldots ,\underbrace{\Gamma u_{j}}_{i_{j}=1},\ldots ,\underbrace{{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}\{u_{r}\}}_{i_{r}=\partial },\ldots )$$. Proof. We first illustrate the proof for the case of a unary monotone map. We show that if $$\delta (f)=(1;1)$$, then $$f_{\sigma }(\Gamma x)=\Gamma (f^{\flat } x)$$. Recall that the $$\sigma$$ extension $$f^{\sigma }:{\mathcal G}_{\psi }(X)\longrightarrow{\mathcal G}_{\psi }(X)$$ of a monotone map f as in equation (1) is defined by instantiating equation (1) in the dual lattice frame of [19] by setting   \begin{align} f^{\sigma}(\Gamma x)=&\,\bigwedge\left\{\alpha_{X}(\,fa)\;|\; a\in{\mathcal L}, \Gamma x\leq\alpha_{X}(a)\right\}=\bigwedge\{\Gamma x_{fa}\;|\;\Gamma x\subseteq\Gamma x_{a}\}\nonumber\\ =&\,\bigwedge\{\Gamma x_{fa}\;|\; a\in x\}=\Gamma\left(\bigvee\{x_{fa}\;|\; a\in x\}\right). \end{align} (14)By (10), $$\,f^{\flat } x=\bigvee \{x_{fa}\;|\; a\in x\}$$, hence $$f^{\sigma }(\Gamma x)=\Gamma (\,f^{\flat } x)$$, qed. Consider now the general case of a distribution type $$\delta =(i_{1},\ldots ,i_{n};1)$$, of output type 1. Literally, the $$\sigma$$-extension defined in [9] is a map $$f^{\sigma }:{\mathcal G}(Z^{i_{1}})\times \cdots \times{\mathcal G}(Z^{i_{n}})\longrightarrow{\mathcal G}_{\psi }(X)$$, where $${\mathcal G}(Z^{i_{j}})=\left\{\begin{smallmatrix} {\!\!\mathcal G}_{\psi }(X) &\textrm{when}\ i_{j}=1\\ {\!\!\mathcal G}_{\phi }(Y) &\textrm{when}\ i_{j}=\partial \end{smallmatrix}\right.$$, defined on closed elements by $$f^{\sigma }(\Gamma u_{1},\ldots ,\Gamma u_{n})=\bigwedge \{\Gamma x_{fa_{1}\cdots a_{n}}\;|\;\bigwedge _{j}(a_{j}\in u_{j})\}$$. Composing with the dual equivalence $$(\psi ,\phi )$$ at the appropriate argument places we get the equivalent definition of the map $$f^{\sigma }:{\mathcal G}_{\psi }(X)^{i_{1}}\times \cdots \times{\mathcal G}_{\psi }(X)^{i_{n}}\longrightarrow{\mathcal G}_{\psi }(X)$$, where $${\mathcal G}_{\psi }(X)^{i_{j}}=\left\{\begin{smallmatrix} {\!\!\!\!\!\mathcal G}_{\psi }(X) & \textrm{when}\ i_{j}=1\\ {\!\!\mathcal G}_{\psi }(X)^{\partial } & \textrm{when}\ i_{j}=\partial \end{smallmatrix}\right.$$, defined by $$f^{\sigma }(\ldots ,\underbrace{\Gamma u_{j}}_{i_{j}=1},\ldots ,\underbrace{{}{^{{\mathop{=}^{\kern-5pt\shortmid}}}}\{u_{r}\}}_{i_{r}=\partial },\ldots )=\bigwedge \{\Gamma x_{fa_{1}\cdots a_{n}}\;|\;\bigwedge _{j}(a_{j}\in u_{j})\}=\Gamma (\bigvee \{x_{fa_{1}\cdots a_{n}}\;|\;\bigwedge _{j}(a_{j}\in u_{j})\})$$. Hence we obtain $$f^{\sigma }(\ldots ,\underbrace{\Gamma u_{j}}_{i_{j}=1},\ldots ,\underbrace{{}{^{{\mathop{=}^{\kern-5pt\shortmid}}}}\{u_{r}\}}_{i_{r}=\partial },\ldots )=\Gamma (f^{\flat } (u_{1},\ldots , u_{n}))$$. We may then conclude that $$\,{\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta }(\ldots ,\underbrace{\Gamma u_{j}}_{i_{j}=1},\ldots ,\underbrace{{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}\{u_{r}\}}_{i_{r}=\partial },\ldots )=\Gamma (f^{\flat }_{\delta }(u_{1},\ldots ,u_{n}))=f^{\sigma }(\ldots ,\underbrace{\Gamma u_{j}}_{i_{j}=1},\ldots,\underbrace{{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}\{u_{r}\}}_{i_{r}=\partial },\ldots )$$. Use now Corollary 3.7, for the particular case of the canonical frame, where it was shown that   $${\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta}(A_{1},\ldots,A_{n})=\bigvee^{i_{j}=1,u_{j}\in A_{j}}_{i_{r}=\partial,A_{r}{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}} u_{r}}{\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta}(\ldots,\underbrace{\Gamma u_{j}}_{i_{j}=1},\ldots,\underbrace{{}^{{\mathop{=}^{\kern-5pt\shortmid}}}\{u_{r}\}}_{i_{r}=\partial},\ldots)$$together with the fact that the $$\sigma$$-extension of f is defined on all stable sets in [9] using join-density of closed elements as precisely the join displayed above to conclude that $$f^{\sigma }_{\delta }={\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta }$$ on $${\mathcal G}_{\psi }(X)$$. For the reader’s benefit, we display the calculation.   \begin{align*} {\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta}(A_{1},\ldots,A_{n}) &=\psi\phi\widehat{\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta}(A_{1},\ldots,A_{n})\\ &=\psi\phi\left(\{x\in X\;|\;\exists u_{1}\cdots u_{n}\;(xR_{\delta} u_{1}\cdots u_{n}\;\wedge\;\bigwedge_{i_{j}=1}(u_{j}\in A_{j}) \;\wedge\;\bigwedge_{i_{r}=\partial}(A_{r}\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; u_{r})) \}\right)\\ &=\psi\phi\left( \bigcup^{i_{j}=1,u_{j}\in A_{j}}_{i_{r}=\partial,A_{r}\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; u_{r}}\{x\in X\;|\; xR_{\delta} u_{1}\cdots u_{n}\}\right)=\psi\phi\left( \bigcup^{i_{j}=1,u_{j}\in A_{j}}_{i_{r}=\partial,A_{r}\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; u_{r}}\Gamma(\,f^{\flat}_{\delta}(u_{1},\ldots,u_{n}))\right)\\ &=\bigvee^{i_{j}=1,u_{j}\in A_{j}}_{i_{r}=\partial,A_{r}\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; u_{r}}\psi\phi\Gamma\left(\,f^{\flat}_{\delta}(u_{1},\ldots,u_{n})\right)=\bigvee^{i_{j}=1,u_{j}\in A_{j}}_{i_{r}=\partial,A_{r}\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; u_{r}}\Gamma\left(\,f^{\flat}_{\delta}(u_{1},\ldots,u_{n})\right)\\ &=\bigvee^{i_{j}=1,u_{j}\in A_{j}}_{i_{r}=\partial,A_{r}\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; u_{r}}f^{\sigma}_{\delta}(\ldots,\underbrace{\Gamma u_{j}}_{i_{j}=1},\ldots, \underbrace{{}{{}^{{\mathop{=}^{\kern-5pt\shortmid}}}}\{u_{r}\}}_{i_{r}=\partial},\ldots)= f^{\sigma}_{\delta}(A_{1},\ldots,A_{n}). \end{align*}This proves the first subclaim of claim 3). For the second subclaim of 3), that $$\ominus _{\delta ^{\prime }}=h^{\sigma }_{\delta ^{\prime }}$$, we just dualize the argument. Indeed, if $$h_{\delta ^{\prime }}:{\mathcal L}^{i_{1}}\times \cdots \times{\mathcal L}^{i_{n}}\longrightarrow{\mathcal L}^{\partial }$$, let $${\mathcal M}={\mathcal L}^{\partial }$$ so that $$h_{\delta ^{\prime }}:{\mathcal M}^{\overline{i_{1}}}\times \cdots \times{\mathcal M}^{\overline{i_{n}}}\longrightarrow{\mathcal M}$$, where $$\overline{i_{j}}$$ is the dual value, i.e. ∂ if $$i_{j}=1$$ and 1 if $$i_{j}=\partial$$. Then h appears as a map of the dual distribution type $$\overline{\delta ^{\prime }}$$ on $${\mathcal M}$$, therefore of output type 1. Hence the previous argument can be dualized, interchanging $${\mathcal G}_{\phi }(Y)$$ for $${\mathcal G}_{\psi }(X)$$, open for closed etc. Writing $$h^{\sigma ,\partial }_{\delta ^{\prime }}$$ for the dual $$\sigma$$-extension of h and composing with the dual equivalence $$\psi ,\phi$$ we obtain what is called the $$\pi$$-extension of h in [9]. We illustrate this for a unary map below.   \begin{align*} \psi\left(\,f_{\sigma}^{\partial}(\Gamma y)\right) =&\,\psi\left(\bigwedge\{\Gamma y_{fa}\;|\; a\in y\}\right) =\bigvee\{\psi\Gamma y_{fa}\;|\; a\in y\} =\bigvee\{\Gamma x_{fa}\;|\; a\in y\} \\ =&\, \bigvee\{\Gamma x_{fa}\;|\; y_{a}\leq y\} =\bigvee\{\Gamma x_{fa}\;|\; \Gamma y\subseteq \Gamma y_{a}\} =\bigvee\{\Gamma x_{fa}\;|\; \psi\Gamma y_{a}\subseteq \psi\Gamma y\} \\ =&\,\bigvee\{\Gamma x_{fa}\;|\;{}{^{{\mathop{=}^{\kern-5pt\shortmid}}}} \{y_{a}\}\subseteq{}{{}^{{\mathop{=}^{\kern-5pt\shortmid}}}} \{y\}\} =\bigvee\{\Gamma x_{fa}\;|\; \alpha_{Y}(a)\subseteq{}{^{{\mathop{=}^{\kern-5pt\shortmid}}}} \{y\}\} =f_{\pi}({}{^{{\mathop{=}^{\kern-5pt\shortmid}}}}\{y\}) \end{align*}The above argument has shown that $$\ominus _{\delta ^{\prime }}=h^{\pi }_{\delta ^{\prime }}$$. By [9], Lemmas 4.3, 4.4, 4.6, the following hold for a unary monotone map f (1) The $$\sigma$$ and $$\pi$$ extensions $$f^{\sigma }, f^{\pi }$$ of f agree on closed or open elements. (2) If either $$f^{\sigma }$$ preserves all joins, or $$f^{\pi }$$ preserves all meets, then $$f^{\sigma }= f^{\pi }$$. (3) If f preserves binary joins then $$f^{\sigma }$$ preserves all joins and if f preserves binary meets, then $$f^{\pi }$$ preserves all meets. Note that the above hold for any map since e.g. an antitone map from $$\mathcal L$$ to $$\mathcal K$$ is a monotone map from $$\mathcal L$$ to $${\mathcal K}^{\partial }$$ and an n-ary map can be regarded as a unary map whose domain is the product lattice. Therefore, since the operators of interest in this article are assumed to be normal, $$\sigma$$ and $$\pi$$-extensions coincide and this concludes the proof of claim 3). To prove claim 4) we essentially only need to verify the following. Lemma 4.3 If g is a normal lattice operator of some distribution type $$\delta$$, then its $$\sigma$$-extension $$g^{\sigma }$$ is a completely normal operator of the same distribution type. Proof. This can be established in one of two ways. Indeed, we may appeal to [12], Theorem 6.6. Alternatively, we may appeal to [9], Lemma 4.6. Either way, the claim of the Lemma is established. We may now complete the proof of Theorem 4.1, showing that the canonical frame of a lattice expansion is a $$\tau$$-frame. We have already pointed out that the frame axioms (F1–F5) clearly hold for the canonical frame, as do axioms (R1–R4). Axioms (R5, R6) hold trivially since the dual relations are defined so as to satisfy the axioms. The frame axioms (R7, R8) were included in order to prove that the generated operators $${\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta },\ominus _{\delta ^{\prime }}$$ have the requisite distribution properties and they are merely a rephrasing of particular instances of (co)distribution. Hence they hold in the canonical frame, following from the fact that the operators generated by the relations were shown to have the complete (co)distribution properties for the distribution types $$\delta ,{\delta ^{\prime }}$$, respectively. Finally, axioms (R9, R10) are equivalent to the claim that the operators $${\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta },\ominus _{\delta ^{\prime }}$$ are the $$\sigma$$-extensions of their restriction to clopen elements and the proof of the latter was the content of claim 3), proven above. This establishes that the dual frame of a normal lattice expansion is a $$\tau$$-frame and it completes the proof of Theorem 4.1. We have then defined, by the above, the object part of functors $$\hat{C}:\textbf{L}_{\tau }\longrightarrow{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}\textbf{Frm}_{\tau }$$, sending a lattice expansion of similarity type $$\tau$$ to its dual frame, and $$\hat{F}:{}{^{{\mathop{=}^{\kern-5pt\shortmid}}}}\textbf{Frm}_{\tau }\longrightarrow \textbf{L}_{\tau }$$, sending a frame to its dual algebra of clopens. Lemma 4.4 $$\hat{C}:\textbf{L}_{\tau }\longrightarrow{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}\textbf{Frm}_{\tau }$$, sending a lattice expansion of similarity type $$\tau$$ to its dual frame, and $$\hat{F}:{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}\textbf{Frm}_{\tau }\longrightarrow \textbf{L}_{\tau }$$, sending a frame to its dual algebra of clopens extend to contravariant functors. Proof. The proof is an extension of the corresponding proof for the underlying categories of lattices and lattice frames given in [19]. By the results of [19], given a homomorphism $$f:{\mathcal A}\longrightarrow{\mathcal B}$$ of $$\tau$$-algebras we obtain a lattice frame morphism $$\hat{F}(f)=(f_{1},f_{2}):(X_{B},{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}} _{B},Y_{B})=\hat{F}({\mathcal B})\longrightarrow \hat{F}({\mathcal A})=(X_{A},{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}} _{A},Y_{A})$$ where $$f_{1}(x)=f^{-1}[x]$$ and $$f_{2}(y)=f^{-1}[y]$$ such that their inverses $$f_{1}^{\ast }=f_{1}^{-1}:X^{\ast }_{A}\longrightarrow X^{\ast }_{B}$$, $$f_{2}^{\ast }=f_{2}^{-1}:Y^{\ast }_{A}\longrightarrow Y^{\ast }_{B}$$ are lattice homomorphisms (where $$X^{\ast }={\mathcal G}_{\kappa o}(X), Y^{\ast }={\mathcal G}_{\kappa o}(Y)$$) satisfying $$f_{1}^{\ast }(\alpha _{A}(a))=\alpha _{B}(b)$$ iff $$f_{2}^{\ast }(\beta _{A}(a))=\beta _{B}(b)$$ iff f(a) = b, where $$\alpha ,\beta$$ are the (co)representation maps, $$\alpha (e)=\{x\in X\;|\; e\in x\}$$ and $$\beta (e)=\{y\in Y\;|\; e\in y\}$$ (with the corresponding subscripts A, B). For a distribution type $$\delta$$, let $$\oplus _{\delta ,A}:{\mathcal A}^{n}\longrightarrow{\mathcal A}$$ and $$\oplus _{\delta ,B}:{\mathcal B}^{n}\longrightarrow{\mathcal B}$$ be the corresponding operators in $${\mathcal A},{\mathcal B}$$, respectively, and $$\oplus _{\delta ,A}^{\sigma }:(X^{\ast }_{A})^{n}\longrightarrow X^{\ast }_{A}$$, $$\oplus _{\delta ,B}^{\sigma }:(X^{\ast }_{B})^{n}\longrightarrow X^{\ast }_{B}$$ their respective representations (their $$\sigma$$-extensions, restricted to clopen sets). We verify that $$f^{\ast }_{1}(\oplus _{\delta ,A}^{\sigma }(A_{1},\ldots ,A_{n}))=\oplus _{\delta ,B}^{\sigma }(f^{\ast }_{1}(A_{1}),\ldots ,f^{\ast }_{1}(A_{n}))$$. To this purpose, let $$A_{i}=\alpha _{A}(a_{i})$$, so that we have   \begin{array}{lll} f^{\ast}_{1}(\oplus_{\delta,A}^{\sigma}(A_{1},\ldots,A_{n})){\hskip-8pt}&=f^{\ast}_{1}(\oplus_{\delta,A}^{\sigma}(\alpha_{A}(a_{1}),\ldots,\alpha_{A}(a_{n}))) & \textrm{by}\ A_{i}=\alpha_{A}(a_{i})\\ &=f^{\ast}_{1}(\alpha_{A}(\oplus_{\delta,A}(a_{1},\ldots,a_{n}))) & \sigma \text{-extensions represent the operators}\\ &=\alpha_{B}(\,f(\oplus_{\delta,A}(a_{1},\ldots,a_{n})))& \textrm{by definition of}\ f^{\ast}_{1} \ \textrm{and results of}\ (19)\\ &=\alpha_{B}(\oplus_{\delta,B}(\,f(a_{1}),\ldots,f(a_{n}))) & f:{\mathcal A}\longrightarrow{\mathcal B}\ \textrm{is a homomorphism}\\ &=\oplus^{\sigma}_{\delta,B}(\alpha_{B}(\,f(a_{1})),\ldots,\alpha_{B}(\,f(a_{n}))) & \sigma \text{-extensions represent the operators}\\ &=\oplus^{\sigma}_{\delta,B}(\,f^{\ast}_{1}(\alpha_{A}(a_{1})),\ldots,f^{\ast}_{1}(\alpha_{A}(a_{n}))) & \textrm{by definition of}\ f^{\ast}_{1} \ \textrm{and results of}\ (19)\\ &=\oplus^{\sigma}_{\delta,B}(\,f^{\ast}_{1}(A_{1}),\ldots,f^{\ast}_{1}(A_{n})) & \textrm{by}\ A_{i}=\alpha_{A}(a_{i}) \end{array}Hence $$f^{\ast }_{1}$$ is a homomorphism and then $$\hat{F}$$ is fully defined as a contravariant functor from L$$_{\tau }$$ to $${ }^{{\mathop{=}^{\kern-5pt\shortmid}}}\textbf{Frm}_{\tau }$$. By [19], Proposition 2.11, every lattice frame (X, $${\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}$$, Y) is the dual of a lattice $${\mathcal L}$$, i.e. it can be concretely viewed as a frame consisting of the sets X, Y of filters and ideals of $$\mathcal L$$ and where the Galois relation $${\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}$$ is the relation x$${\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}$$y iff $$x\cap y\neq \emptyset$$. Furthermore, by [19], Proposition 2.11 again, every lattice frame morphism $$(\,f_{1},f_{2}):(X_{B},{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}} _{B},Y_{B})\longrightarrow (X_{A},{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}} _{A},Y_{A})$$ arises from a unique lattice homomorphism $$f:{\mathcal A}\longrightarrow{\mathcal B}$$, where f(a) = b iff $$\,f^{\ast }_{1}(\alpha _{A}(a))=\alpha _{B}(b)$$ and then $$f_{1}(x)=f^{-1}[x]$$ and $$f^{\ast }_{1}=f^{-1}_{1}$$. Given a morphism $$(\,f_{1},f_{2}):(X_{B},{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}} _{B},Y_{B})\longrightarrow (X_{A},{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}} _{A},Y_{A})$$ of $$\tau$$-frames, let $$(X_{B},{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}} _{B},Y_{B})=\hat{F}({\mathcal B})$$, $$(X_{A},{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}} _{A},Y_{A})=\hat{F}({\mathcal A})$$, let also $$(\,f_{1},f_{2})=\hat{F}(\,f)$$, where $$f:{\mathcal A}\longrightarrow{\mathcal B}$$ is the unique homomorphism of the underlying lattices of the $$\tau$$-algebras $${\mathcal A,B}$$ and $$f^{\ast }_{1}:X^{\ast }_{A}\longrightarrow X^{\ast }_{B}$$ be defined as detailed above. We verify below that $$f:{\mathcal A}\longrightarrow{\mathcal B}$$ is a $$\tau$$-algebra homomorphism, i.e. $$f(\oplus _{\delta ,A}(a_{1},\ldots ,a_{n}))=\oplus _{\delta ,B}(\,f(a_{1}),\ldots ,f(a_{n}))$$. By definition of $$\tau$$-frame morphisms (Definition 3.9), $$f^{\ast }_{1}$$ is a $$\tau$$-algebra homomorphism and we then have   \begin{align*} &f(\oplus_{\delta,A}(a_{1},\ldots,a_{n}))=b\\ &\begin{array}{llll} {\hskip-8pt}& \textrm{iff} & f^{\ast}_{1}(\alpha_{A}(\oplus_{\delta,A}(a_{1},\ldots,a_{n})))=\alpha_{B}(b) & \textrm{by definition of f} \\ & \textrm{iff} & f^{\ast}_{1}(\oplus^{\sigma}_{\delta,A}(\alpha_{A}(a_{1}),\ldots,\alpha_{A}(a_{n}))=\alpha_{B}(b) & \textrm{by representation} \\ & \textrm{iff} & \oplus^{\sigma}_{\delta,B}(\,f^{\ast}_{1}(\alpha_{A}(a_{1})),\ldots,f^{\ast}_{1}(\alpha_{A}(a_{n})))=\alpha_{B}(b) & f^{\ast}_{1} \ \textrm{is a homomorphism} \\ & \textrm{iff} & \oplus^{\sigma}_{\delta,B}(\alpha_{B}(\,f(a_{1})),\ldots,\alpha_{B}(\,f(a_{n})))=\alpha_{B}(b)& \textrm{by definition of f} \\ & \textrm{iff} & \alpha_{B}(\oplus_{\delta,B}(\,f(a_{1}),\ldots,f(a_{n})))=\alpha_{B}(b) & \textrm{by representation} \\ & \textrm{iff} & \alpha_{B}(\oplus_{\delta,B}(\,f(a_{1}),\ldots,f(a_{n})))=\alpha_{B}(\,f(\oplus_{\delta,A}(a_{1},\ldots,a_{n}))) & \\ & \textrm{iff} & \oplus_{\delta,B}(\,f(a_{1}),\ldots,f(a_{n}))=f(\oplus_{\delta,A}(a_{1},\ldots,a_{n})) & \textrm{by representation} \end{array} \end{align*}Hence, f is a $$\tau$$-algebra homomorphism and we may define $$\hat{C}(\,f_{1},f_{2})$$ to be the unique $$\tau$$-algebra homomorphism $$f:{\mathcal A}\longrightarrow{\mathcal B}$$, as above, thus extending $$\hat{C}$$ to a contravariant functor from the category of $$\tau$$-frames to the category of $$\tau$$-algebras. Theorem 4.5 (Duality) The contravariant functors $$\hat{F},\hat{C}$$ form a dual equivalence of the categories of normal lattice expansions of similarity type $$\tau$$ with the corresponding categories of $$\tau$$-frames. Proof. For the isomorphism $${\mathcal A}\backsimeq \hat{C}\hat{F}({\mathcal A})=({\mathcal A}_{+})^{+}$$, where $${\mathcal A}_{+}=\hat{F}(A)=(X,{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}} ,Y)$$ is the dual frame of $$\mathcal A$$ and $$X^{\ast }={\mathcal G}_{\kappa o}(X)$$ is the lattice of clopen elements of its canonical extension $${\mathcal G}_{\psi }(X)$$, the results of [19] ensure that $${\mathcal A}, ({\mathcal A}_{+})^{+}$$ are isomorphic as lattices, via the representation map $$\alpha (a)=\{x\in X\;|\; a\in x\}$$. For an n-ary normal lattice operator $$\oplus _{\delta }:{\mathcal A}^{n}\longrightarrow{\mathcal A}$$, where $$\delta \in \tau$$ is of any output type 1, or ∂, it was shown in Proposition 4.1 that $$\alpha (\oplus _{\delta }(a_{1},\ldots ,a_{n}))=\oplus ^{\sigma }_{\delta }(\alpha (a_{1}),\ldots ,\alpha (a_{n}))$$, where the latter is $${\bigcirc}{\kern-6.4pt\mid}\ \,^{\sigma }_{\delta }(\alpha (a_{1}),\ldots ,\alpha (a_{n}))$$, if $$\delta \in \tau _{1}$$ and it is $$\ominus ^{\sigma }_{\delta }(\alpha (a_{1}),\ldots ,\alpha (a_{n}))$$ when $$\delta \in \tau _{\partial }$$, hence the lattice isomorphism $$\alpha$$ is a homomorphism of $$\tau$$-algebras, given also Corollary 3.1. Therefore $${\mathcal A}\backsimeq \hat{C}\hat{F}({\mathcal A})=({\mathcal A}_{+})^{+}$$. For the isomorphism $$\mathfrak{F}\backsimeq \hat{F}\hat{C}(\mathfrak{F})=(\mathfrak{F}^{+})_{+}$$, where $$\mathfrak{F}=(X,{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}} ,Y)$$ and by the results of [19] and by the arguments in the proof of Lemma 4.4, we may as well assume that $$\mathfrak{F}=\hat{F}({\mathcal A})$$, for a unique $$\tau$$-algebra $$\mathcal A$$ such that $${\mathcal A}\backsimeq X^{\ast }$$, where recall that the latter is the $$\tau$$-algebra of clopens of $${\mathcal G}_{\psi }(X)$$. Therefore, $$\hat{F}\hat{C}(\mathfrak{F})=\hat{F}\hat{C}\hat{F}({\mathcal A})\backsimeq \hat{F}({\mathcal A})=\mathfrak{F}$$, given that $$\hat{F},\hat{C}$$ form an adjunction, by the duality of [19]. 5 Modeling full BCK Ono [26–29] studied a number of systems arising from the Gentzen system LJ for intuitionistic logic by dropping a combination of the structural rules of exchange, contraction and weakening (perhaps also association) and expanding the logical signature of the language to include the operator symbols ∘ (fusion, cotenability), $$\leftarrow$$ (reverse implication) and a constant $$\mathfrak{t}$$. The algebraic semantics of these systems has been investigated by Hiroakira Ono, see [29], and others. Following Ono, we let FL be the system with all structural rules dropped, which is precisely the (associative) Full Lambek calculus, and for r ⊆{c, e, w} we designate by $$\textbf{FL}_{r}$$ the system resulting by adding to FL the structural rules in r (where c abbreviates ‘contraction’, e abbreviates ‘exchange’ and similarly for w and ‘weakening’). With the exception of $$\textbf{FL}_{ecw}$$, which is precisely LJ, distribution of conjunctions over disjunctions and conversely does not hold, unless explicitly postulated in the axiomatisation. An FL-algebra is a structure $$\langle L,\leq ,\wedge ,\vee ,0,1,\leftarrow ,\circ ,\rightarrow ,\mathfrak{t}\rangle$$ where (1) ⟨L, ≤, ∧, ∨, 0, 1⟩ is a bounded lattice. (2) $$\langle L,\leq ,\circ ,\mathfrak{t}\rangle$$ is a partially-ordered monoid (∘ is monotone and associative and $$\mathfrak{t}$$ is a two-sided identity element $$a\circ \mathfrak{t}=a=\mathfrak{t}\circ a$$). (3) $$\leftarrow ,\circ ,\rightarrow$$ are residuated, i.e. a ∘ b ≤ c iff $$b\leq a\rightarrow c$$ iff $$a\leq c\leftarrow b$$. (4) for any a ∈ L, a ∘ 0 = 0 = 0 ∘ a. An FL-algebra is known as a residuated lattice. FL-algebras (residuated lattices) are precisely the algebraic models of the (associative) full Lambek calculus. An FL$$_{ew}$$-algebra adds to the axiomatisation the exchange (commutativity) axiom a ∘ b = b ∘ a for the cotenability operator (in which case $$\leftarrow$$ and $$\rightarrow$$ coincide), as well as the weakening axiom b ∘ a ≤ a, in which case combining with commutativity a ∘ b ≤ a ∧ b follows. In addition, by $$1\circ \mathfrak{t}\leq \mathfrak{t}$$, the identity $$\mathfrak{t}=1$$ holds in FL$$_{ew}$$-algebras. FL$$_{ew}$$-algebras are also referred to in the literature as full BCK-algebras, corresponding to full BCK-logic, resulting from BCK whose purely implicational signature is expanded to include conjunction and disjunction connectives, alongside the cotenability logical operator and the constants 0,1. Algebraically, they constitute the class of commutative integral residuated lattices. The language of full BCK is displayed below, where P is a nonempty, countable set of propositional variables.   $$L\ni\varphi\;:=\; p\;(p\in P)\;|\;\top\;|\;\bot\;|\; \varphi\wedge\varphi\;|\;\varphi\vee\varphi\;|\;\varphi\circ\varphi\;|\;\varphi{\rightarrow}\varphi.$$Since we have no interest in this article in studying proof theoretic issues, we may as well assume that the proof system is presented as a symmetric consequence system $$\varphi \vdash \psi$$, directly encoding the corresponding algebraic specification (and thereby being sound and complete in the class of commutative integral residuated lattices). Both operators $$\circ ,\rightarrow$$ are normal, with respective distribution types $$\delta (\circ )=(1,1;1)=\delta$$ and $$\delta (\rightarrow )=(1,\partial ;\partial )={\delta ^{\prime}}$$. This section provides an application of the proposed representation and duality results, modeling the full BCK calculus. BCK-frames are the appropriate $$\tau$$-frames of Section 3, detailed in the sequel, but where we drop, as usual, the topology (axiom F4). There is an option, as pointed out in the Introduction (repeating a point we first made in [20]), as far as interpretations are concerned. The first option considers plain Kripke frames, though two-sorted, and interprets a propositional variable as any stable set V (p) in the full complex algebra $${\mathcal G}_{\psi }(X)$$ of the frame and co-interprets it as a co-stable set in $${\mathcal G}_{\phi }(Y)$$, under the restriction that the co-interpretation is the Galois-dual of the interpretation: $$V{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}(p)=V(p){ }^{{\mathop{=}^{\kern-5pt\shortmid}}}$$. The second option considers general frames and restricts (co)interpretations to assign, respectively, a closed set to a propositional variable. In the general frames approach the received interpretation of the logical operators is typically verbatim the same as in the distributive case, which may be of significant interests in some applied contexts. 5.1 Frames and soundness Definition 5.1 (BCK Frames) A BCK-frame is a $$\tau$$-frame (except for dropping the topology, axiom F4) $$\mathfrak{F}=(X,(R_{\circ },R^{\partial }_{\circ }),{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}} ,Y,(S_>,S^{\partial }_>))$$, for $$\tau =(\delta ,{\delta ^{\prime }})$$, with two additional conditions (E) and (Res). Explicitly, (1) X, Y are nonempty sets and $${\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}$$ ⊆ X × Y is a binary relation generating a Galois connection defined on U ⊆ X and V ⊆ Y by   \begin{align*} \phi(U)=&\,U^{{\mathop{=}^{\kern-5pt\shortmid}}} =\{y\in Y\;|\;\forall u\in U\; u{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}} y\}=\{y\in Y\;|\; U\ {\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\ y\} \\ \psi(V)=&\,{}^{{\mathop{=}^{\kern-5pt\shortmid}}} V =\{x\in X\;|\; \forall v\in V\; x{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}} v\}=\{x\in X\;|\; x\ {\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\ V\} \end{align*}and we let $${\mathcal G}_{\psi }(X),{\mathcal G}_{\phi }(Y)$$ be the complete lattices of stable, $$A=\psi \phi A$$, and co-stable, $$B=\phi \psi B$$, subsets of X and Y, respectively. (F1) The relations x ≤ z iff $$\{x\}^{{\mathop{=}^{\kern-5pt\shortmid}}}\subseteq \{z\}^{{\mathop{=}^{\kern-5pt\shortmid}}}$$ on X and y ≤ v iff $${ }^{{\mathop{=}^{\kern-5pt\shortmid}}}\{y\}\subseteq{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}\{z\}$$ on Y are partial orders on X and of Y, respectively. (F2) $${\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}$$ is increasing in each argument place, i.e. x$${\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}$$y, x ≤ z, y ≤ v imply z$${\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}$$v. (F3) Clopen sets are closed under finite intersections and closed sets are closed under arbitrary intersections, for each of $${\mathcal G}_{\psi }(X),{\mathcal G}_{\phi }(Y)$$ (see Definition 2.3 for the definition of closed, open and clopen elements). (F5) The family of closed sets is the meet-closure of the family of clopen sets, for each of $${\mathcal G}_{\psi }(X),{\mathcal G}_{\phi }(Y)$$. (R1) $$R_{\circ }\;\subseteq X\times X\times X$$ and $$S^{\partial }_>\;\subseteq Y\times X\times Y$$. (R2) $$R_{\circ }, S^{\partial }_>$$ are increasing in the first argument place and decreasing in every other argument place. (R3) For any $$(u_{1},u_{2})\in (X\times X)$$, the set $$R_{\circ } u_{1} u_{2}$$ is a closed element of $${\mathcal G}_{\psi }(X)$$ (in other words, there exists a point z ∈ X such that for any x ∈ X, $$xR_{\circ } u_{1} u_{2}$$ iff z ≤ x) and if both $$u_{i}$$ are clopen, then so is $$R_{\circ } u_{1} u_{2}$$. (R4) For any $$(u_{1},u_{2})\in (X\times Y)$$, the set $$S^{\partial }_> u_{1}u_{2}$$ is a closed element of $${\mathcal G}_{\phi }(Y)$$ (i.e. there is a point v ∈ Y such that for all y ∈ Y, $$yS^{\partial }_> u_{1} u_{2}$$ iff v ≤ y) and if both $$u_{i}$$ are clopen, then so is $$S^{\partial }_> u_{1} u_{2}$$. (R5) $$R^{\partial }_{\circ }\subseteq Y\times (X\times X)$$ and $$\forall (u_{1}, u_{2})\in (X\times X)\;R^{\partial }_{\circ } u_{1} u_{2}\;=\;(R_{\circ } u_{1} u_{2}){ }^{{\mathop{=}^{\kern-5pt\shortmid}}}$$. (R6) $$S_>\subseteq X\times (X\times Y)$$ and $$\forall (v_{1},v_{2})\in (X\times Y)\;S_> v_{1}v_{2}\;=\;{}{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}(S^{\partial }_> v_{1}v_{2})$$. (R7) For all x ∈ X and all $$u_{s},u^{\prime }_{s}\in X$$, for s = 1, 2,   \begin{align*} xR_{\circ} u_{1} (u_{2}\cap u^{\prime}_{2}) &\longrightarrow \forall y\in Y\;(\,yR^{\partial}_{\circ} u_{1} u_{2}\;\wedge\;yR^{\partial}_{\circ} u_{1} u^{\prime}_{2}\;\longrightarrow\;x\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; y) \\ xR_{\circ}(u_{1}\cap u^{\prime}_{1})u_{2} &\longrightarrow \forall y\in Y\;(\,yR^{\partial}_{\circ} u_{1} u_{2}\;\wedge\;yR^{\partial}_{\circ} u^{\prime}_{1} u_{2}\;\longrightarrow\;x\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; y). \end{align*} (R8) For all $$y,u_{2},u^{\prime }_{2}\in Y, u_{1},u^{\prime }_{1}\in X$$  \begin{align*} yS^{\partial}_> u_{1}\left(u_{2}\cap u^{\prime}_{2}\right) &\longrightarrow \forall x\in X\;\left(\,yS_> u_{1} u_{2}\;\wedge\;yS_> u_{1}\cdots u^{\prime}_{2}\;\longrightarrow\;x\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; y\right) \\ yS^{\partial}_> \left(u_{1}\cap u^{\prime}_{1}\right) u_{2}&\longrightarrow\forall x\in X\;\left(\,yS_> u_{1} u_{2}\;\wedge\;yS_{\delta} u^{\prime}_{1} u_{2}\;\longrightarrow\;x\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; y\right). \end{align*} (R9) For all $$(u_{1},u_{2})\in X\times X$$ and z ∈ X, if for all clopen elements $$u^{s}_{*}$$ below $$u_{s}$$, for each s = 1, 2, there exist $$(v_{1},v_{2})\in X\times X$$ such that $$zR_{\delta } v_{1}v_{2}$$ and $$u^{s}_{*}\leq v_{s}$$, for each s = 1, 2, then there exist $$(v^{\prime }_{1},v^{\prime }_{2})\in X\times X$$ such that $$zR_{\delta } v^{\prime }_{1} v^{\prime }_{2}$$ and $$u_{s}\leq v^{\prime }_{s}$$, for each s = 1, 2. (R10) For all $$(u_{1},u_{2})\in X\times Y$$ and y ∈ Y, if for all clopen elements $$u^{s}_{*}$$ below $$u_{s}$$, for each s = 1, 2, there exist $$(v_{1},v_{2})\in X\times Y$$ such that $$yS^{\partial }_{\delta ^{\prime }} v_{1} v_{2}$$ and $$u^{s}_{*}\leq v_{s}$$, for each s = 1, 2, then there exist $$(v^{\prime }_{1},v^{\prime }_{2})\in X\times Y$$ such that $$yS^{\partial }_{\delta ^{\prime }} v^{\prime }_{1} v^{\prime }_{2}$$ and $$u_{s}\leq v^{\prime }_{s}$$, for each s = 1, 2. (E) For all u, x, z ∈ X, $$\;uR_{\circ } xz\ \textrm{iff}\; uR_{\circ } zx$$. (Res) The following are equivalent, where $$A,B,C\in{\mathcal G}_{\psi }(X)$$ are stable sets $$\forall u\in X\;\forall x\in A\;\forall z\in B\;(uR_{\circ } xz\;\Longrightarrow \;u\in C)$$ $$\forall z\in B\;\forall y\in Y\;\forall v\in X\;\forall w\in Y\;(yS^{\partial }_> vw\;\wedge \;v\in A\;\wedge \;C{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}} w\;\Longrightarrow \;z{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}} y)$$. Note that we have not listed axiom (R8) for $$\tau$$-frames as it refers to distribution types the kind of which does not occur in the similarity type $$\tau$$ under consideration. Definition 5.2 For $$U,U^{\prime }\subseteq X$$, $$A,A^{\prime }\in{\mathcal G}_{\psi }(X)$$ and $$V,V^{\prime }\subseteq Y$$, $$B,B^{\prime }\in{\mathcal G}_{\phi }(Y)$$ define the operators   $${\hskip-10pt}\widehat{\bigcirc}{\kern-6.4pt\mid}\ \,_{\circ}(U,U^{\prime}) = \{x\in X\;|\;\exists u,u^{\prime}\in X\;(xR_{\circ} uu^{\prime}\;\wedge\;u\in U\;\wedge\;u^{\prime}\in U^{\prime})\}$$ (15)  $${\hskip-160pt}{\bigcirc}{\kern-6.4pt\mid}\ \,_{\circ}(A,A^{\prime}) = \psi\phi\widehat{\bigcirc}{\kern-6.4pt\mid}\ \,_{\circ}(A,A^{\prime})$$ (16)  $${\hskip-150pt}{\bigcirc}{\kern-6.4pt\mid}\ \,^{\partial}_{\circ}(B,B^{\prime}) = \phi{\bigcirc}{\kern-6.4pt\mid}\ \,_{\circ}(\psi B,\psi B^{\prime})$$ (17)  \begin{align} \widehat{\ominus}^{\partial}_>(V,V^{\prime}) = \{y\in Y\;|\; \exists u\in X\;\exists v\in Y\;(yS^{\partial}_>uv\;\wedge\;u\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; V\;\wedge\; v\in V^{\prime})\} \end{align} (18)  $${\hskip-160pt}\ominus^{\partial}_>(B,B^{\prime}) = \phi\psi\widehat{\ominus}^{\partial}_>(B,B^{\prime})$$ (19)  $${\hskip-145pt}\ominus_>(A,A^{\prime}) = \psi\ominus^{\partial}_>(\phi A,\phi A^{\prime}).$$ (20)Simplify notation by writing $$A\,{\bigcirc}{\kern-6.4pt\mid}\ \,\, A^{\prime }$$, $$B\,{\bigcirc}{\kern-6.4pt\mid}\ \,^{\partial } B^{\prime }$$ for $${\bigcirc}{\kern-6.4pt\mid}\ \,_{\circ }(A,A^{\prime }), {\bigcirc}{\kern-6.4pt\mid}\ \,^{\partial }_{\circ }(B,B^{\prime })$$, respectively. Similarly, write $$A\Rightarrow A^{\prime }, B\Rightarrow ^{\partial } B^{\prime }$$ for $$\ominus _>(A,A^{\prime }), \ominus ^{\partial }_>(B,B^{\prime })$$ respectively, recalling that $$B \ {\bigcirc}{\kern-6.4pt\mid}\ \,^{\partial } B^{\prime }=\phi (\psi B\,{\bigcirc}{\kern-6.4pt\mid}\ \,\, \psi B^{\prime })$$ while also $$A\Rightarrow A^{\prime }=\psi (\phi A\Rightarrow ^{\partial }\phi A^{\prime })$$. In the following lemma we prove properties of the operators that will be used in the sequel. Lemma 5.3 By the results of Section 3 the following hold. (1) For any x, z ∈ X, y ∈ Y,   \begin{align*} \widehat{\bigcirc}{\kern-6.4pt\mid}\ \,_{\circ}(\Gamma x,\Gamma z)=&\,(\Gamma x){\bigcirc}{\kern-6.4pt\mid}\ \,(\Gamma z) =R_{\circ} xz\in{\mathcal G}_{\kappa}(X)\\ \widehat{\ominus}^{\partial}_>\left(\{x\}{{}^{{\mathop{=}^{\kern-5pt\shortmid}}}},\Gamma y\right)=&\,\left(\{x\}{{}^{{\mathop{=}^{\kern-5pt\shortmid}}}}\right)\Rightarrow^{\partial}(\Gamma y)=S^{\partial}_>xy\in{\mathcal G}_{\kappa}(Y). \end{align*} (2) The operators $$\widehat{\bigcirc}{\kern-6.4pt\mid}\ \,_{\circ }, \widehat{\ominus }_>$$ restrict to operators on clopen elements of $${\mathcal G}_{\psi }(X)$$ and, thereby, so do the operators $${\bigcirc}{\kern-6.4pt\mid}\ \,,\Rightarrow$$. (3) $$B\ {\bigcirc}{\kern-6.4pt\mid}\ \,^{\partial } B^{\prime }=\{y\in Y\;|\;\forall u,u^{\prime }\in X\;(u{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}} B\;\wedge \;u^{\prime }{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; B^{\prime }\;\Longrightarrow \;yR^{\partial }_{\circ } uu^{\prime })\}$$. (4) $$A\Rightarrow A^{\prime } =\{x\in X\;|\;\forall u\in X\;\forall v\in Y\;(u\in A\;\wedge \; A^{\prime }{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}} v\;\Longrightarrow \;xS_>uv)\}$$. (5) $$A{\bigcirc}{\kern-6.4pt\mid}\ \,A^{\prime }=\bigvee ^{x\in A}_{z\in A^{\prime }}(\Gamma x\ {\bigcirc}{\kern-6.4pt\mid}\ \,\ \Gamma z)$$ and, similarly, $$B\Rightarrow ^{\partial } B^{\prime }=\bigvee ^{x{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}} B}_{v\in B^{\prime }}(\{x\}{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}\Rightarrow ^{\partial }\Gamma v)$$. (6) The operators $${\bigcirc}{\kern-6.4pt\mid}\ \,,\Rightarrow$$ are the $$\sigma$$-extensions of their restrictions to clopens and hence they are completely normal operators in the full complex algebra of the frame, of distribution types $$\delta =(1,1;1)$$ and $${\delta ^{\prime }}=(1,\partial ;\partial )$$, respectively. Proof. 1) is an instance of Lemma 3.3 and 2) is one of Corollary 3.4, while 3) and 4) follow by Lemma 3.6 and 5) follows by Corollary 3.7. Recall that only the F axioms and axioms R1–R4 are used in the proofs of Lemmas 3.3, 3.6 and Corollaries 3.4, 3.7 are consequences of Lemma 3.3. Recall also that the definitions of the dual relations were introduced in the proof of Lemma 3.6 and were subsequently included as frame axioms (R5, R6) in Section 3.2. 6) follows by Proposition 3.13, whose proof relied on the frame axioms (R7–R10). Lemma 5.4 The frame conditions (E,Res) ensure commutativity $$A\,{\bigcirc}{\kern-6.4pt\mid}\ \,\, A^{\prime }=A^{\prime }\,{\bigcirc}{\kern-6.4pt\mid}\ \,\, A$$ and residuation $$A\,{\bigcirc}{\kern-6.4pt\mid}\ \,\, B\subseteq C$$ iff B ⊆ A ⇒ C. Proof. By Lemma 5.3, $$A\ {\bigcirc}{\kern-6.4pt\mid}\ \,\ A^{\prime }=\bigvee ^{x\in A}_{z\in A^{\prime }}(\Gamma x\ {\bigcirc}{\kern-6.4pt\mid}\ \,\ \Gamma z)=\bigvee ^{x\in A}_{z\in A^{\prime }}R_{\circ } xz =\bigvee ^{x\in A}_{z\in A^{\prime }}R_{\circ } zx=\bigvee ^{x\in A}_{z\in A^{\prime }}(\Gamma z\ {\bigcirc}{\kern-6.4pt\mid}\ \,\ \Gamma x)=A^{\prime }\ {\bigcirc}{\kern-6.4pt\mid}\ \,\ A$$. For residuation, $$A\,{\bigcirc}{\kern-6.4pt\mid}\ \,\, B\subseteq C$$ iff $$\,\bigvee ^{x\in A}_{z\in B}R_{\circ } xz\subseteq C$$ iff $$\forall x\in A\;\forall z\in B\; R_{\circ } xz\subseteq C$$. Note also that   \begin{align*} A\Rightarrow C&= \psi(\phi A\Rightarrow^{\partial}\phi C)\\ &=\psi\phi\psi\{\,y\in Y\;|\;\exists v\in X\exists w\in Y\;(yS^{\partial}_>vw\;\wedge\;v\in A\;\wedge\;C\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; w)\}\\ &=\psi\{\,y\in Y\;|\;\exists v\in X\exists w\in Y\;(yS^{\partial}_>vw\;\wedge\;v\in A\;\wedge\;C\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; w)\} \end{align*}hence   \begin{align*} B\subseteq A\Rightarrow C &\textrm{iff}\ \ B\subseteq \psi\{\,y\in Y\;|\;\exists v\in X\exists w\in Y\;(yS^{\partial}_>vw\;\wedge\;v\in A\;\wedge\;C\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; w)\}\\ &\textrm{iff}\ \ B \ {\mathop{=}^{\kern-5pt\raise-4pt\shortmid}} \ \{\,y\in Y\;|\;\exists v\in X\exists w\in Y\;(yS^{\partial}_>vw\;\wedge\;v\in A\;\wedge\;C\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; w)\}\\ &\textrm{iff}\ \ \forall z\in B\;\forall y\in Y\;(\exists v\in X\exists w\in Y\;(\,yS^{\partial}_>vw\;\wedge\;v\in A\;\wedge\;C\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; w)\;\Longrightarrow\;z\;{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}\; y). \end{align*}Therefore, the frame condition (Res) is equivalent to the residuation condition for $${\bigcirc}{\kern-6.4pt\mid}\ \,,\Rightarrow$$. 5.2 Models, soundness and completeness BCK models $$\mathfrak{M}=(\mathfrak{F},V,V{ }^{{\mathop{=}^{\kern-5pt\shortmid}}})$$ consist of a BCK-frame $$\mathfrak{F}=(X,(R_{\circ },R^{\partial }_{\circ }),{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}} ,Y,(S_>,S^{\partial }_>))$$ together with an interpretation V and a co-interpretation $$V{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}$$ assigning to a propositional variable a stable and a co-stable set, respectively, satisfying $$V{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}(p)=V(p){ }^{{\mathop{=}^{\kern-5pt\shortmid}}}$$. As pointed out in [20], there appear to exist two options in defining models and in each case soundness and completeness can be established. According to the first option, V assigns any stable set $$V(p)\in{\mathcal G}_{\psi }(X)$$ to a propositional variable p, but the resulting interpretation clauses are non-standard (see for example [8] where the fusion-implication fragment of FL is modeled). Alternatively, we may restrict interpretations to assign a closed set $$V(p)\in{\mathcal G}_{\kappa }(X)\subseteq{\mathcal G}_{\psi }(X)$$, in which case the interpretation pattern follows lines familiar from the distributive case. The difference then is one of working with plain (two-sorted) Kripke frames, in which case the intended interpretation of some operators may be lost, or working with general frames (restricting interpretations to assign closed elements to propositional variables), in which case the intended meaning of operators is re-captured. The relational representation of operators and subsequent Stone duality we have presented can be used in each of the above cases and, for explicitness, we present both approaches in modeling BCK. 5.2.1 Plain (two-sorted) Kripke frames and models Let $$\mathfrak{M}=(\mathfrak{F},V,V{ }^{{\mathop{=}^{\kern-5pt\shortmid}}})$$ be a frame $$\mathfrak{F}=(X,R_{\circ },S_>,{\mathop{=}^{\kern-5pt\raise-4pt\shortmid}} ,R^{\partial }_{\circ },S^{\partial }_>,Y)$$ together with an interpretation V assigning a stable set $$V(p)\in{\mathcal G}_{\psi }(X)$$ and a co-interpretation $$V{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}$$ assigning a costable set $$V{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}(p)\in{\mathcal G}_{\phi }(Y)$$ such that $$V{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}(p)=V(p){ }^{{\mathop{=}^{\kern-5pt\shortmid}}}$$. Extend the interpretation and co-interpretation recursively to all sentences as in Table 1, where R is the complement of the Galois relation $${\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}$$ of the frame. Table 1 Interpretation and co-interpretation of full BCK in plain Kripke frames $$[\![p]\!]$$  = V (p)    $$[\![\top]\!]$$  = X    $$[\![\bot]\!]$$  $$=\varnothing _{\psi }$$  $$=\bigcap{\mathcal G}_{\psi }(X)$$  $$[\![\vartheta \wedge \chi ]\!]$$  $$=[\![\vartheta ]\!]\cap [\![\chi ]\!]$$    $$[\![\vartheta \vee \chi ]\!]$$  $$=\psi \phi ([\![\vartheta ]\!]\cup [\![\chi ]\!])$$  $$=\{x\in X\;|\; \forall y\in Y\;(xRy\;\longrightarrow y\not \in (\!|\vartheta |\!)\;\vee \;y\not \in (\!|\chi |\!))\}$$  $$[\![\vartheta \circ \chi ]\!]$$  $$=[\![\vartheta ]\!]{\bigcirc}{\kern-6.4pt\mid}\ \,[\![\chi ]\!]$$  $$=\psi \phi \{x\in X\;|\;\exists z,z^{\prime }\in X\;(xR_{\circ } zz^{\prime }\;\wedge \;z\in [\![\vartheta ]\!]\;\wedge \;z^{\prime }\in [\![\chi ]\!])\}$$  $$[\![\vartheta \rightarrow \chi ]\!]$$  $$=[\![\vartheta ]\!]\Rightarrow [\![\chi ]\!]$$  $$=\{x\in X\;|\; \forall z\in X\;\forall y\in Y;(z\in [\![\vartheta ]\!]\;\wedge \;y\in (\!|\chi |\!) \;\longrightarrow \;xS_>zy)\}$$  $$(\!|p|\!)$$  $$=[\![p]\!]{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}$$    $$(\!|\top|\!)$$  $$=\varnothing _{\phi }$$  $$=\;\bigcap{\mathcal G}_{\phi }(Y)$$  $$(\!|\bot|\!)$$  = Y    $$(\!|\vartheta \wedge \chi |\!)$$  $$=\phi \psi ((\!|\vartheta |\!) \cup (\!|\chi |\!) )$$  $$=\;\{y\in Y\;|\;\forall x\in X\;(xRy\;\longrightarrow \;x\not \in [\![\vartheta ]\!]\;\vee \;x\not \in [\![\chi ]\!])\}$$  $$(\!|\vartheta \vee \chi |\!)$$  $$=(\!|\vartheta |\!)\cap (\!|\chi |\!)$$    $$(\!|\vartheta \circ \chi |\!)$$  $$=(\!|\vartheta |\!) {\bigcirc}{\kern-6.4pt\mid}\ \,^{\partial }(\!|\chi |\!)$$  $$=\{y\in Y\;|\;\forall x,z\in X\;(x\in [\![\vartheta ]\!]\;\wedge \;z\in [\![\chi ]\!]\;\longrightarrow \; yR^{\partial }_{\circ } xz)\}$$  $$(\!|\vartheta \rightarrow \chi |\!)$$  $$=(\!|\vartheta |\!) \Rightarrow ^{\partial }(\!|\chi |\!)$$  $$=\;\phi \psi \{y\in Y\;|\;\exists x\in X\exists v\in Y\; (yS^{\partial }_> xv\;\wedge \;x\in [\![\vartheta ]\!]\;\wedge \;v\in (\!|\chi |\!) )\}$$  $$[\![p]\!]$$  = V (p)    $$[\![\top]\!]$$  = X    $$[\![\bot]\!]$$  $$=\varnothing _{\psi }$$  $$=\bigcap{\mathcal G}_{\psi }(X)$$  $$[\![\vartheta \wedge \chi ]\!]$$  $$=[\![\vartheta ]\!]\cap [\![\chi ]\!]$$    $$[\![\vartheta \vee \chi ]\!]$$  $$=\psi \phi ([\![\vartheta ]\!]\cup [\![\chi ]\!])$$  $$=\{x\in X\;|\; \forall y\in Y\;(xRy\;\longrightarrow y\not \in (\!|\vartheta |\!)\;\vee \;y\not \in (\!|\chi |\!))\}$$  $$[\![\vartheta \circ \chi ]\!]$$  $$=[\![\vartheta ]\!]{\bigcirc}{\kern-6.4pt\mid}\ \,[\![\chi ]\!]$$  $$=\psi \phi \{x\in X\;|\;\exists z,z^{\prime }\in X\;(xR_{\circ } zz^{\prime }\;\wedge \;z\in [\![\vartheta ]\!]\;\wedge \;z^{\prime }\in [\![\chi ]\!])\}$$  $$[\![\vartheta \rightarrow \chi ]\!]$$  $$=[\![\vartheta ]\!]\Rightarrow [\![\chi ]\!]$$  $$=\{x\in X\;|\; \forall z\in X\;\forall y\in Y;(z\in [\![\vartheta ]\!]\;\wedge \;y\in (\!|\chi |\!) \;\longrightarrow \;xS_>zy)\}$$  $$(\!|p|\!)$$  $$=[\![p]\!]{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}$$    $$(\!|\top|\!)$$  $$=\varnothing _{\phi }$$  $$=\;\bigcap{\mathcal G}_{\phi }(Y)$$  $$(\!|\bot|\!)$$  = Y    $$(\!|\vartheta \wedge \chi |\!)$$  $$=\phi \psi ((\!|\vartheta |\!) \cup (\!|\chi |\!) )$$  $$=\;\{y\in Y\;|\;\forall x\in X\;(xRy\;\longrightarrow \;x\not \in [\![\vartheta ]\!]\;\vee \;x\not \in [\![\chi ]\!])\}$$  $$(\!|\vartheta \vee \chi |\!)$$  $$=(\!|\vartheta |\!)\cap (\!|\chi |\!)$$    $$(\!|\vartheta \circ \chi |\!)$$  $$=(\!|\vartheta |\!) {\bigcirc}{\kern-6.4pt\mid}\ \,^{\partial }(\!|\chi |\!)$$  $$=\{y\in Y\;|\;\forall x,z\in X\;(x\in [\![\vartheta ]\!]\;\wedge \;z\in [\![\chi ]\!]\;\longrightarrow \; yR^{\partial }_{\circ } xz)\}$$  $$(\!|\vartheta \rightarrow \chi |\!)$$  $$=(\!|\vartheta |\!) \Rightarrow ^{\partial }(\!|\chi |\!)$$  $$=\;\phi \psi \{y\in Y\;|\;\exists x\in X\exists v\in Y\; (yS^{\partial }_> xv\;\wedge \;x\in [\![\vartheta ]\!]\;\wedge \;v\in (\!|\chi |\!) )\}$$  Table 1 Interpretation and co-interpretation of full BCK in plain Kripke frames $$[\![p]\!]$$  = V (p)    $$[\![\top]\!]$$  = X    $$[\![\bot]\!]$$  $$=\varnothing _{\psi }$$  $$=\bigcap{\mathcal G}_{\psi }(X)$$  $$[\![\vartheta \wedge \chi ]\!]$$  $$=[\![\vartheta ]\!]\cap [\![\chi ]\!]$$    $$[\![\vartheta \vee \chi ]\!]$$  $$=\psi \phi ([\![\vartheta ]\!]\cup [\![\chi ]\!])$$  $$=\{x\in X\;|\; \forall y\in Y\;(xRy\;\longrightarrow y\not \in (\!|\vartheta |\!)\;\vee \;y\not \in (\!|\chi |\!))\}$$  $$[\![\vartheta \circ \chi ]\!]$$  $$=[\![\vartheta ]\!]{\bigcirc}{\kern-6.4pt\mid}\ \,[\![\chi ]\!]$$  $$=\psi \phi \{x\in X\;|\;\exists z,z^{\prime }\in X\;(xR_{\circ } zz^{\prime }\;\wedge \;z\in [\![\vartheta ]\!]\;\wedge \;z^{\prime }\in [\![\chi ]\!])\}$$  $$[\![\vartheta \rightarrow \chi ]\!]$$  $$=[\![\vartheta ]\!]\Rightarrow [\![\chi ]\!]$$  $$=\{x\in X\;|\; \forall z\in X\;\forall y\in Y;(z\in [\![\vartheta ]\!]\;\wedge \;y\in (\!|\chi |\!) \;\longrightarrow \;xS_>zy)\}$$  $$(\!|p|\!)$$  $$=[\![p]\!]{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}$$    $$(\!|\top|\!)$$  $$=\varnothing _{\phi }$$  $$=\;\bigcap{\mathcal G}_{\phi }(Y)$$  $$(\!|\bot|\!)$$  = Y    $$(\!|\vartheta \wedge \chi |\!)$$  $$=\phi \psi ((\!|\vartheta |\!) \cup (\!|\chi |\!) )$$  $$=\;\{y\in Y\;|\;\forall x\in X\;(xRy\;\longrightarrow \;x\not \in [\![\vartheta ]\!]\;\vee \;x\not \in [\![\chi ]\!])\}$$  $$(\!|\vartheta \vee \chi |\!)$$  $$=(\!|\vartheta |\!)\cap (\!|\chi |\!)$$    $$(\!|\vartheta \circ \chi |\!)$$  $$=(\!|\vartheta |\!) {\bigcirc}{\kern-6.4pt\mid}\ \,^{\partial }(\!|\chi |\!)$$  $$=\{y\in Y\;|\;\forall x,z\in X\;(x\in [\![\vartheta ]\!]\;\wedge \;z\in [\![\chi ]\!]\;\longrightarrow \; yR^{\partial }_{\circ } xz)\}$$  $$(\!|\vartheta \rightarrow \chi |\!)$$  $$=(\!|\vartheta |\!) \Rightarrow ^{\partial }(\!|\chi |\!)$$  $$=\;\phi \psi \{y\in Y\;|\;\exists x\in X\exists v\in Y\; (yS^{\partial }_> xv\;\wedge \;x\in [\![\vartheta ]\!]\;\wedge \;v\in (\!|\chi |\!) )\}$$  $$[\![p]\!]$$  = V (p)    $$[\![\top]\!]$$  = X    $$[\![\bot]\!]$$  $$=\varnothing _{\psi }$$  $$=\bigcap{\mathcal G}_{\psi }(X)$$  $$[\![\vartheta \wedge \chi ]\!]$$  $$=[\![\vartheta ]\!]\cap [\![\chi ]\!]$$    $$[\![\vartheta \vee \chi ]\!]$$  $$=\psi \phi ([\![\vartheta ]\!]\cup [\![\chi ]\!])$$  $$=\{x\in X\;|\; \forall y\in Y\;(xRy\;\longrightarrow y\not \in (\!|\vartheta |\!)\;\vee \;y\not \in (\!|\chi |\!))\}$$  $$[\![\vartheta \circ \chi ]\!]$$  $$=[\![\vartheta ]\!]{\bigcirc}{\kern-6.4pt\mid}\ \,[\![\chi ]\!]$$  $$=\psi \phi \{x\in X\;|\;\exists z,z^{\prime }\in X\;(xR_{\circ } zz^{\prime }\;\wedge \;z\in [\![\vartheta ]\!]\;\wedge \;z^{\prime }\in [\![\chi ]\!])\}$$  $$[\![\vartheta \rightarrow \chi ]\!]$$  $$=[\![\vartheta ]\!]\Rightarrow [\![\chi ]\!]$$  $$=\{x\in X\;|\; \forall z\in X\;\forall y\in Y;(z\in [\![\vartheta ]\!]\;\wedge \;y\in (\!|\chi |\!) \;\longrightarrow \;xS_>zy)\}$$  $$(\!|p|\!)$$  $$=[\![p]\!]{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}$$    $$(\!|\top|\!)$$  $$=\varnothing _{\phi }$$  $$=\;\bigcap{\mathcal G}_{\phi }(Y)$$  $$(\!|\bot|\!)$$  = Y    $$(\!|\vartheta \wedge \chi |\!)$$  $$=\phi \psi ((\!|\vartheta |\!) \cup (\!|\chi |\!) )$$  $$=\;\{y\in Y\;|\;\forall x\in X\;(xRy\;\longrightarrow \;x\not \in [\![\vartheta ]\!]\;\vee \;x\not \in [\![\chi ]\!])\}$$  $$(\!|\vartheta \vee \chi |\!)$$  $$=(\!|\vartheta |\!)\cap (\!|\chi |\!)$$    $$(\!|\vartheta \circ \chi |\!)$$  $$=(\!|\vartheta |\!) {\bigcirc}{\kern-6.4pt\mid}\ \,^{\partial }(\!|\chi |\!)$$  $$=\{y\in Y\;|\;\forall x,z\in X\;(x\in [\![\vartheta ]\!]\;\wedge \;z\in [\![\chi ]\!]\;\longrightarrow \; yR^{\partial }_{\circ } xz)\}$$  $$(\!|\vartheta \rightarrow \chi |\!)$$  $$=(\!|\vartheta |\!) \Rightarrow ^{\partial }(\!|\chi |\!)$$  $$=\;\phi \psi \{y\in Y\;|\;\exists x\in X\exists v\in Y\; (yS^{\partial }_> xv\;\wedge \;x\in [\![\vartheta ]\!]\;\wedge \;v\in (\!|\chi |\!) )\}$$  Let $$\Vdash , \Vdash ^{\!\!\partial }$$ be defined by $$x\Vdash \vartheta$$ iff $$x\in [\![\vartheta ]\!]\in{\mathcal G}_{\psi }(X)$$ and $$y\Vdash ^{\!\!\partial }\vartheta$$ iff $$y\in (\!|\vartheta |\!)\in{\mathcal G}_{\phi }(Y)$$. Unfolding definitions and recalling that R is the complement of the Galois relation $${\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}$$ of the frame we obtain in particular for fusion and implication the following (co)satisfaction clauses, where $$\overline{S}_>$$ is the complement of $$S_>$$ and we let $$\tilde{S}^{\partial }_>=RS^{\partial }_>$$ designate the composition of R with $$S^{\partial }_>$$ and $$\tilde{R}_{\circ }=R^{-1}R_{\circ }$$ stand for the composition of the converse $$R^{-1}$$ of R with $$R_{\circ }$$.   \begin{array}{lll} X\ni u\Vdash\vartheta\circ\chi & \textrm{iff} & \quad \forall y\in Y\;\left(uRy\;\longrightarrow\;\exists z,z^{\prime}\in X\;\left(\,y\tilde{R}_{\circ} zz^{\prime}\;\wedge z\Vdash\vartheta\;\wedge\;z^{\prime}\Vdash\chi\right)\right) \\ Y\ni y\Vdash^{\!\!\partial}\vartheta\circ\chi & \textrm{iff} & \quad \forall x,z\in X\;\left(x\Vdash\vartheta\;\wedge\;z\Vdash\chi\;\longrightarrow\;yR^{\partial}_{\circ} xz\right) \\ X\ni u\Vdash\vartheta\! \rightarrow \!\chi & \textrm{iff} & \quad \forall x\in X\;\forall y\in Y\;(x\Vdash\vartheta\;\wedge\; u\overline{S}_>xy\;\longrightarrow\;y\not\Vdash^{\!\!\partial}\chi) \\ Y\in y\Vdash^{\!\!\partial}\vartheta\! \rightarrow \!\chi & \textrm{iff} & \quad \forall z\in X\;\left(zRy\;\longrightarrow\;\exists x\in X\;\exists v\in Y\;\left(x\Vdash\vartheta\;\wedge\;z\tilde{ S}^{\partial}_>xv\;\wedge\;v\Vdash^{\!\!\partial}\chi\right)\right). \end{array} The interested reader may wish to compare the above clauses with the corresponding clauses in the generalized Kripke frames approach of [8], where fusion and implication are modeled. Satisfiability and validity of a sentence at a state, frame or class of frames is defined as usual. A sequent $$\varphi \vdash \psi$$ is valid, written $$\vartheta \Vdash \chi$$, iff for any model and state x ∈ X in the underlying frame of the model, if $$x\Vdash \vartheta$$, then $$x\Vdash \chi$$, or, equivalently, if $$y\Vdash ^{\!\!\partial }\chi$$, then $$y\Vdash ^{\!\!\partial }\vartheta$$, for any y ∈ Y. Theorem 5.5 (Soundness and completeness) Full BCK is sound and complete in plain, two-sorted Kripke frame semantics. Proof. We have established the distribution properties of $$\, {\bigcirc}{\kern-6.4pt\mid}\ \,$$ and ⇒ in Lemma 5.3 and commutativity of $${\bigcirc}{\kern-6.4pt\mid}\ \,$$ and residuation of $${\bigcirc}{\kern-6.4pt\mid}\ \,$$ with ⇒ have been established in Lemma 5.4. This proves soundness of BCK. Completeness follows from the representation theorem proven in Section 4. 5.2.2 General (two-sorted) frames and models General (two-sorted) frames are like plain two-sorted frames except that they restrict interpretations and co-interpretations to assign closed elements of $${\mathcal G}_{\psi }(X)$$ and $${\mathcal G}_{\phi }(Y)$$, respectively, to propositional variables. The recursive extension to an interpretation [[ ]] and a co-interpretation (| |) is given by the same clauses as in plain frames, except that taking the Galois closure $$\psi \phi$$ in the cases of the interpretation of fusion and $$\phi \psi$$ for the co-interpretation of implication become redundant as the respective sets are already Galois closed (stable). The satisfaction and co-satisfaction (refutation) relations can be specified by the clauses in Table 2. Table 2 Interpretation and co-interpretation of full BCK in general Kripke frames (x, u, z ∈ X, y, v ∈ Y, R = $${\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}$$?, $$\varnothing _{\psi }=\bigcap{\mathcal G}_{\psi }(X),\;\varnothing _{\phi }=\bigcap{\mathcal G}_{\phi }(Y)$$)  $$x\Vdash p$$  iff  x ∈ V (p)  $$y\Vdash ^{\!\!\partial } p$$  iff  $$y\in V{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}(p)$$  $$x\Vdash \top$$    always  $$y\Vdash ^{\!\!\partial }\bot$$    always  $$x\Vdash \bot$$  iff  $$x\in \varnothing _{\psi }$$  $$y\Vdash ^{\!\!\partial }\top$$  iff  $$y\in \varnothing _{\phi }$$  $$x\Vdash \varphi \wedge \psi$$  iff  $$x\Vdash \varphi$$ and $$x\Vdash \psi$$  $$y\Vdash ^{\!\!\partial }\varphi \vee \psi$$  iff  $$y\Vdash ^{\!\!\partial }\varphi$$ and $$y\Vdash ^{\!\!\partial }\psi$$  $$x\Vdash \varphi \vee \psi$$  iff  $$\forall y\;(xRy\;\Longrightarrow \;y\not \Vdash ^{\!\!\partial }\varphi \ \textrm{or}\ y\not \Vdash ^{\!\!\partial }\psi )$$  $$y\Vdash ^{\!\!\partial }\varphi \wedge \psi$$  iff  $$\forall x\;(xRy\;\Longrightarrow \;x\not \Vdash \varphi \ \textrm{or}\ x\not \Vdash \psi )$$  $$x\Vdash \varphi \circ \psi$$  iff  $$\exists u, z\;(xR_{\circ } uz\ \textrm{and}\ u\Vdash \varphi \ \textrm{and}\ z\Vdash \psi )$$  $$y\Vdash ^{\!\!\partial }\varphi \circ \psi$$  iff  $$\forall x,z\;(x\Vdash \varphi \ \textrm{and}\ z\Vdash \psi \;\Longrightarrow \; yR^{\partial }_{\circ } xz)$$  $$x\Vdash \varphi \rightarrow \psi$$  iff  $$\forall z\forall y\;(z\Vdash \varphi \ \textrm{and}\ y\Vdash ^{\!\!\partial }\psi \;\Longrightarrow \;xS_>zy)$$  $$y\Vdash ^{\!\!\partial }\varphi \rightarrow \psi$$  iff  $$\exists x, v\;(yR^{\partial }_>xv\ \textrm{and}\ x\Vdash \varphi \ \textrm{but}\ v\Vdash ^{\!\!\partial }\psi )$$  (x, u, z ∈ X, y, v ∈ Y, R = $${\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}$$?, $$\varnothing _{\psi }=\bigcap{\mathcal G}_{\psi }(X),\;\varnothing _{\phi }=\bigcap{\mathcal G}_{\phi }(Y)$$)  $$x\Vdash p$$  iff  x ∈ V (p)  $$y\Vdash ^{\!\!\partial } p$$  iff  $$y\in V{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}(p)$$  $$x\Vdash \top$$    always  $$y\Vdash ^{\!\!\partial }\bot$$    always  $$x\Vdash \bot$$  iff  $$x\in \varnothing _{\psi }$$  $$y\Vdash ^{\!\!\partial }\top$$  iff  $$y\in \varnothing _{\phi }$$  $$x\Vdash \varphi \wedge \psi$$  iff  $$x\Vdash \varphi$$ and $$x\Vdash \psi$$  $$y\Vdash ^{\!\!\partial }\varphi \vee \psi$$  iff  $$y\Vdash ^{\!\!\partial }\varphi$$ and $$y\Vdash ^{\!\!\partial }\psi$$  $$x\Vdash \varphi \vee \psi$$  iff  $$\forall y\;(xRy\;\Longrightarrow \;y\not \Vdash ^{\!\!\partial }\varphi \ \textrm{or}\ y\not \Vdash ^{\!\!\partial }\psi )$$  $$y\Vdash ^{\!\!\partial }\varphi \wedge \psi$$  iff  $$\forall x\;(xRy\;\Longrightarrow \;x\not \Vdash \varphi \ \textrm{or}\ x\not \Vdash \psi )$$  $$x\Vdash \varphi \circ \psi$$  iff  $$\exists u, z\;(xR_{\circ } uz\ \textrm{and}\ u\Vdash \varphi \ \textrm{and}\ z\Vdash \psi )$$  $$y\Vdash ^{\!\!\partial }\varphi \circ \psi$$  iff  $$\forall x,z\;(x\Vdash \varphi \ \textrm{and}\ z\Vdash \psi \;\Longrightarrow \; yR^{\partial }_{\circ } xz)$$  $$x\Vdash \varphi \rightarrow \psi$$  iff  $$\forall z\forall y\;(z\Vdash \varphi \ \textrm{and}\ y\Vdash ^{\!\!\partial }\psi \;\Longrightarrow \;xS_>zy)$$  $$y\Vdash ^{\!\!\partial }\varphi \rightarrow \psi$$  iff  $$\exists x, v\;(yR^{\partial }_>xv\ \textrm{and}\ x\Vdash \varphi \ \textrm{but}\ v\Vdash ^{\!\!\partial }\psi )$$  Table 2 Interpretation and co-interpretation of full BCK in general Kripke frames (x, u, z ∈ X, y, v ∈ Y, R = $${\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}$$?, $$\varnothing _{\psi }=\bigcap{\mathcal G}_{\psi }(X),\;\varnothing _{\phi }=\bigcap{\mathcal G}_{\phi }(Y)$$)  $$x\Vdash p$$  iff  x ∈ V (p)  $$y\Vdash ^{\!\!\partial } p$$  iff  $$y\in V{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}(p)$$  $$x\Vdash \top$$    always  $$y\Vdash ^{\!\!\partial }\bot$$    always  $$x\Vdash \bot$$  iff  $$x\in \varnothing _{\psi }$$  $$y\Vdash ^{\!\!\partial }\top$$  iff  $$y\in \varnothing _{\phi }$$  $$x\Vdash \varphi \wedge \psi$$  iff  $$x\Vdash \varphi$$ and $$x\Vdash \psi$$  $$y\Vdash ^{\!\!\partial }\varphi \vee \psi$$  iff  $$y\Vdash ^{\!\!\partial }\varphi$$ and $$y\Vdash ^{\!\!\partial }\psi$$  $$x\Vdash \varphi \vee \psi$$  iff  $$\forall y\;(xRy\;\Longrightarrow \;y\not \Vdash ^{\!\!\partial }\varphi \ \textrm{or}\ y\not \Vdash ^{\!\!\partial }\psi )$$  $$y\Vdash ^{\!\!\partial }\varphi \wedge \psi$$  iff  $$\forall x\;(xRy\;\Longrightarrow \;x\not \Vdash \varphi \ \textrm{or}\ x\not \Vdash \psi )$$  $$x\Vdash \varphi \circ \psi$$  iff  $$\exists u, z\;(xR_{\circ } uz\ \textrm{and}\ u\Vdash \varphi \ \textrm{and}\ z\Vdash \psi )$$  $$y\Vdash ^{\!\!\partial }\varphi \circ \psi$$  iff  $$\forall x,z\;(x\Vdash \varphi \ \textrm{and}\ z\Vdash \psi \;\Longrightarrow \; yR^{\partial }_{\circ } xz)$$  $$x\Vdash \varphi \rightarrow \psi$$  iff  $$\forall z\forall y\;(z\Vdash \varphi \ \textrm{and}\ y\Vdash ^{\!\!\partial }\psi \;\Longrightarrow \;xS_>zy)$$  $$y\Vdash ^{\!\!\partial }\varphi \rightarrow \psi$$  iff  $$\exists x, v\;(yR^{\partial }_>xv\ \textrm{and}\ x\Vdash \varphi \ \textrm{but}\ v\Vdash ^{\!\!\partial }\psi )$$  (x, u, z ∈ X, y, v ∈ Y, R = $${\mathop{=}^{\kern-5pt\raise-4pt\shortmid}}$$?, $$\varnothing _{\psi }=\bigcap{\mathcal G}_{\psi }(X),\;\varnothing _{\phi }=\bigcap{\mathcal G}_{\phi }(Y)$$)  $$x\Vdash p$$  iff  x ∈ V (p)  $$y\Vdash ^{\!\!\partial } p$$  iff  $$y\in V{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}(p)$$  $$x\Vdash \top$$    always  $$y\Vdash ^{\!\!\partial }\bot$$    always  $$x\Vdash \bot$$  iff  $$x\in \varnothing _{\psi }$$  $$y\Vdash ^{\!\!\partial }\top$$  iff  $$y\in \varnothing _{\phi }$$  $$x\Vdash \varphi \wedge \psi$$  iff  $$x\Vdash \varphi$$ and $$x\Vdash \psi$$  $$y\Vdash ^{\!\!\partial }\varphi \vee \psi$$  iff  $$y\Vdash ^{\!\!\partial }\varphi$$ and $$y\Vdash ^{\!\!\partial }\psi$$  $$x\Vdash \varphi \vee \psi$$  iff  $$\forall y\;(xRy\;\Longrightarrow \;y\not \Vdash ^{\!\!\partial }\varphi \ \textrm{or}\ y\not \Vdash ^{\!\!\partial }\psi )$$  $$y\Vdash ^{\!\!\partial }\varphi \wedge \psi$$  iff  $$\forall x\;(xRy\;\Longrightarrow \;x\not \Vdash \varphi \ \textrm{or}\ x\not \Vdash \psi )$$  $$x\Vdash \varphi \circ \psi$$  iff  $$\exists u, z\;(xR_{\circ } uz\ \textrm{and}\ u\Vdash \varphi \ \textrm{and}\ z\Vdash \psi )$$  $$y\Vdash ^{\!\!\partial }\varphi \circ \psi$$  iff  $$\forall x,z\;(x\Vdash \varphi \ \textrm{and}\ z\Vdash \psi \;\Longrightarrow \; yR^{\partial }_{\circ } xz)$$  $$x\Vdash \varphi \rightarrow \psi$$  iff  $$\forall z\forall y\;(z\Vdash \varphi \ \textrm{and}\ y\Vdash ^{\!\!\partial }\psi \;\Longrightarrow \;xS_>zy)$$  $$y\Vdash ^{\!\!\partial }\varphi \rightarrow \psi$$  iff  $$\exists x, v\;(yR^{\partial }_>xv\ \textrm{and}\ x\Vdash \varphi \ \textrm{but}\ v\Vdash ^{\!\!\partial }\psi )$$  Proposition 5.6 (Soundness) Full BCK is sound in the class of general frames defined. Proof. Letting $$[\![\varphi ]\!]=\{x\in X\;|\; x\Vdash \varphi \}$$ and $$(\!|\varphi |\!) =\{y\in Y\;|\; y\Vdash ^{\!\!\partial }\varphi \}$$, the reader can easily verify that conjunctions and disjunctions are interpreted as intersections and closures of unions, respectively, while $$[\![\varphi \circ \psi ]\!]=[\![\varphi ]\!]\ {\bigcirc}{\kern-6.4pt\mid}\ \,\ [\![\psi ]\!]$$, $$[\![\varphi \rightarrow \psi ]\!]=[\![\varphi ]\!]\Rightarrow [\![\psi ]\!]$$ and, dually, $$(\!|\varphi \circ \psi |\!) =(\!|\varphi |\!)\ {\bigcirc}{\kern-6.4pt\mid}\ \,^{\partial }(\!|\psi |\!)$$ and $$(\!|\varphi \rightarrow \psi |\!) =(\!|\varphi |\!) \Rightarrow ^{\partial }(\!|\psi |\!)$$. This follows from the fact that the restriction that $$V{ }^{{\mathop{=}^{\kern-5pt\shortmid}}}(p)=V(p){ }^{{\mathop{=}^{\kern-5pt\shortmid}}}$$ effectively imposes that propositional variables are (co)interpreted as clopen sets. Soundness then follows by Lemmas 5.3, 5.4. Theorem 5.7 (Completeness) Full BCK is complete in the class of general frames defined. Proof. Completeness follows immediately by the representation argument presented in Section 4, with arguments instantiated to the distribution types of interest for BCK. The canonical representation map indeed sends the equivalence class $$a=[\vartheta]$$ of a sentence $$\vartheta$$ to the clopen set $$\Gamma x_{a}=\{x\in X\;|\; a\in x\}$$ and, similarly for the co-representation map. Details can be safely left to the interested reader. 6 Conclusions We have shown in this article that the duality of the categories of bounded lattices and ⊥-frames proven in [19] lifts to a duality of the categories of normal lattice expan-sions of similarity type $$\tau$$ with the corresponding categories of $$\tau$$-frames. A lattice operator, of some distribution type $$\delta$$, is represented as a set operator $${\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta }$$ or $$\ominus _{\delta }$$, depending on the output type of $$\delta$$, canonically generated by a relation, while the restriction of $${\bigcirc}{\kern-6.4pt\mid}\ \,_{\delta }/\ominus _{\delta }$$ on closed sets (of its domain) is the $$\sigma$$-extension of its restriction on clopens, extended to all (co)stable sets using join-density of closed sets. Extensions of normal lattice operators in canonical extensions have been algebraically constructed in [9] and, in the context of a Stone type duality, in [24, 25]. The contribution of the present article lies in the fact that it presents a relational representation of the operators which can be applied to provide relational (Kripke-style) semantics to various logical calculi lacking the axiom of distribution of conjunction over disjunction and conversely. The present article constitutes an abstraction and generalisation over [20] (http://rgdoi.net/10.13140/RG.2.2.34134.55362/1), an article by this author treating the semantics of modal logic over an implicative, non-distributive lattice with an intuitionistic type of negation. A first version of the current article was made public under the title ‘Relational representation of operators in canonical extensions of normal lattice expansions’ [21]. To illustrate the approach and keep this article self-contained, we have included an application in Section 5, modeling full BCK. In as far as the semantics of non-distributive logical calculi is concerned we repeat here our conclusion from [20] that there appears to exist a choice to be made, namely between pursuing a uniform algebraic approach based on canonical extensions and then abandoning the standard interpretation of e.g. boxes and diamonds, or taking a more applied stance and preferring to abandon uniformity of approach when it comes to semantic issues. The choice boils down to either (a) considering all interpretations assigning just any stable set to propositional variables as admissible, but then the received interpretation of familiar operators must be abandoned, or (b) we may opt for recapturing the familiar meaning of operators despite the absence of distribution, but then interpretations must be restricted to the closed ones, assigning a closed element to a propositional variable. The present article presents the necessary duality theory in support of either one of the above choices. 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For permissions, please e-mail: journals.permission@oup.com. This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/about_us/legal/notices) For permissions, please e-mail: journals. permissions@oup.com

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