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Lower and Upper Concept Lattices

Lower and Upper Concept Lattices Formal Concept Analysis is a mathematical theory of data analysis using formal contexts and concept lattices. In this paper, two new types of concept lattices are introduced by using notions from domain theory (in particular, Hoare and Smyth powerdomains). Based on a Galois connection, we prove the fundamental theorem of the Formal Concept Analysis, as well as other properties of lower and upper formal concepts. In this way, we provide new models to represent and retrieve the information in data and knowledge systems. Mathematics Subject Classification 2010: 03G10, 06A15, 68P20. Key words: concept lattice, Galois connection, powerdomain, data analysis. 1. Introduction Formal Concept Analysis (FCA) is a mathematical theory which analyze the relationships between a set of objects and the set of their properties (attributes) and offers a way to organize, to build and to represent conceptual structures. As a result of investigating and processing the given information the FCA methods build a collection of concepts which are the natural clusters of the objects, according to their attributes. This set of formal concepts may be partially ordered, thus a system of sub- and superconcepts, a conceptual hierarchy, can be obtained. Methods to discover and to analyze dependencies between data and to vizualize these informations are also provided by this approach. All these features make FCA a powerful tool which has been succesfully applied in a wide range of application fields. In the last decade, FCA has been providing a support structure for information retrieval, web search, data mining, gene expression data analysis, latent semantic indexing. Formal concept analysis was proposed by Wille [12] as a mathematical method of data analysis. This approach takes a binary relation between a set of objects and a set of attributes, which are properties of the objects and creates a space of "concepts". Giving to these sets a concrete meaning, one can get different ways to model the information; for emple, one can consider the objects as documents (for emple web pages), while the attributes are terms. A natural idea which was first developed by Burusco and FuentesGonzales [4] was to consider fuzzy contexts in which one assigns to every object a truth degree to which the object has an attribute. They proved that the set of fuzzy concepts is a complete lattice and give a constructive method to build this lattice. However it has been proven that some of the properties from the classical formal concept theory do not hold in this new setting. Pollandt [11] and Belohlavek [2] independently introduced another type of formal concept based on a context with values in a complete residuated lattice, which proved to be a feasibile approach. Based on a Galois connection they proved that all important results from the classical theory of Wille still hold in their setting. In order to reduce the size of the concept lattice which corresponds to large contexts Belohlavek et al. [3] proposed a new approach which generalizes both the fuzzy concept lattice and the lattice of crisply generated fuzzy concepts. Thus, to reduce the complexity of the concept lattice they used as parameters two unary functions defined on the residuated lattice called hedges. Another method to choose the significant data from a very large context was introduced by Zhang et al. [13], which defined three kinds of variable threshold concept lattices. In [8] Krajci constructed the so-called generalized concept lattice. He defined the notion of concept by considering a three-sorted structure of truth degrees which does not include the implication operators. Krajci showed that every concept lattice with hedges is isomorphic to some generalized concept lattice. Another interesting approach was proposed by Georgescu and Popescu [7] which developed the theory of rough concept lattices in a fuzzy setting. In all the articles we have previously mentioned the notion of formal concept was defined as a fixed point of a Galois connection between two partially ordered spaces of sets. Both in classical and fuzzy formal contexts these spaces of sets were endowed with the inclusion order. In our article we define two new types of derivation operators which form a Galois connection between the powerset of objects and the powerset of attributes but endowed with two different types of orderings. On the powerset of attributes we consider the Hoare and, respectively, the Smyth preorders, thus getting the Hoare (lower) and, respectively, the Smyth (upper) powerdomains. Based on the Galois connection and on notions from domain theory, we define two new types of formal concepts, namely the lower and the upper formal concepts and study the structure of the set of concepts. In this new setting we prove the fundamental theorem of the FCA, which states that the set of concepts is a complete lattice, as well as other properties of lower and upper formal concepts. These new types of conceptual structures provide new models to cluster, analyze and represent the information in databases. It is known that concept lattices are mostly used for IR applications, to improve the search strategies like query refinement, ranking or document classification. In these tasks, a query submitted by a user is considered as a concept intent, while the documents retrieved by the system are assimilated with its extent. In our approach the hierarchy in the concept lattices is derived from the Hoare and, respectively, the Smyth preorders and, consequently, it is different from the conceptual structures noticed above. This results in different approximations for a query (concept intent) addressed to a search system as in the classical setting. The paper is organized as it follows. In Section 2 we briefly overview the classical Formal Concept Analysis results and some basic notions about the Hoare and Smyth preorders. Then we introduce the notions of lower and upper formal concept and study their properties in Sections 3 and, respectively 4. An emple for the theoretical model is presented in Section 5. 2. Preliminaries 2.1. Basics of formal concept analysis Definition 1. A formal context is a triple (X, Y, R) with X,Y being abstract sets and R X × Y a relation between X and Y . The elements of X are called objects, those of Y attributes and R the incidence relation of the context (X, Y, R). If xRy we say that "the object x has the attribute y" ( or "y is an attribute of object x"). We define now the derivation operators: Definition 2. Let (X, Y, R) be a formal context. The applications : (P (X) , ) (P (Y ) , ) and : (P (Y ) , ) (P (X) , ) defined by (A) = {y Y | x A, xRy} , A = , () = Y (B) = {x X | y B, xRy} , B = , () = X are called the derivation operators. The set (A) Y, denotes the set of attributes common to all objects in A, while (B) denotes the set of objects which have every attribute in B. It is known that the derivation operators , form an antitone Galois connection between the posets (P (X) , ) and (P (Y ) , ). Definition 3. Let (X, Y, R) be a formal context. A pair (A, B) P (X) × P (Y ) is said to be a formal concept of (X, Y, R) if (A) = B and (B) = A. The sets A, B are called the extent and, respectively, the intent of the formal concept (A, B).The set of all formal concepts is denoted by C (X, Y, R) . If (A, B) is a formal concept, then the sets A, B are maximal according to the properties stated in the definitions. Every formal concept can be obtained as a pair ( ((A)) , (A)), A P (X) or as a pair ( (B) , ( (B))), B P (Y ) . On the set C (X, Y, R) a partial order is been defined; thus the concept (A1 , B1 ) is said to be less than the concept (A2 , B2 ) if A1 A2 . The fundamental theorem of FCA states that the set of formal concepts is a complete lattice: Theorem 4. The set C (X, Y, R) endowed with the relation is a complete lattice which is called concept lattice or Galois lattice. This lattice is denoted by C(X, Y, R). The proof of this result can be found in [6]. 2.2. Hoare and Smyth preorders In this subsection, we consider a partially ordered set (X, ) and we denote by Pf (X) the collection of finite subsets of X. If A X we denote by A, (respectively by A), the lower (upper) closure of the set A i.e. the sets A = {x X|a A, x a}, A = {x X|a A, a x}. We remember the Hoare and Smyth preorderings on Pf (X), which will be important in our study. Definition 5. For all A, B Pf (X) we define the relations l and u on the powerset Pf (X) by A l B if a A, b B, such that a b, and A u B if b B, a A such that a b. The preorders (Pf (X) , l ) and (Pf (X) , u ) are called the Hoare (or lower) and, respectively, the Smyth (or upper) powerdomains of (X, ). It is wellknown that powerdomains have been used for defining a semantics for the programming languages with non-determinism, but in our study we use them as a mathematical support for approximation in partially ordered sets. We may write B l A rather than A l B and =l if A l B and B l A. Similar denotations are used for the upper preorder. Proposition 6. Let (X, ) be a partially ordered set. For all A, B Pf (X) the next equivalences hold: i.A ii.A lB uB if and only if A B if and only if A B Definition 7. Let f : (X, ) (Y, ) and g : (Y, ) (X, ) be two maps between two preordered sets (X, ) , (Y, ). The pair (f, g) is called an antitone Galois connection if 1. for all x1 , x2 X, x1 x2 , f (x1 ) f (x2 ) (f is antitone); 2. for all y1 , y2 Y , y1 y2 , g (y1 ) g (y2 ) (g is antitone); 3. for all x X, x g (f (x)) ( g f is extensive); 4. for all y Y, y f (g (y)) ( f g is extensive). The two maps are called dually adjoint to each other. The above definition is slightly different than the usual one, because (X, ) , (Y, ) are only preordered sets. We notice that the results we are interested to are still true in this case. 3. Lower formal concepts In what follows we consider X a set of objects and Y a finite set of properties (attributes) of objects. In Information Retrieval and Data Mining applications one often consider X to be a set of documents (web pages, for emple), while Y is a hierarchy of terms describing the documents in the form of a thesaurus. Thus it is natural to endow the set of attributes Y with a partial order which can be interpreted as a specialization/generalization relation between the elements of Y. The relation y1 y2 means that the attribute y1 specializes the attribute y2 or that y2 generalizes y1 . Definition 8. An attribute ordered formal context is a triple (X, (Y, ), R) with R X × Y being a relation between X and Y . We call R the incidence relation of the context (X, (Y, ) , R). If xRy we say that "the object x has the attribute y" ( or "y is an attribute of object x"). If x X, we denote the set {y Y | xRy} by R (x) . Using a different order on the powerset of attributes we introduce a new type of formal concept and study the properties of the corresponding concept lattice. First, we define two new derivation operators type: Definition 9. Let (X, (Y, ) , R) be an attribute ordered formal context. The operators l : (P (X) , ) (P (Y ) , l ) and l : (P (Y ) , l ) (P (X) , ) defined by: l (A) = inf {R (x) |x A} , A X, A = and l () = Y, l the l (B) = {x X|B l R (x)} , B P (Y ) , are called lower derivation operators. We denoted by inf obtained in P (Y ) considering the preorder l . infimum Remark 10. By proposition 6 (i) we have l (A) = R (x) , hence y l (A) means that for all x A, x has an attribute y such that y y. An object x is in l (B) if for every y B, x has an attribute y such that y y. Theorem 11. The pair (l , l ) form an antitone Galois connection between the powersets (P (X) , ) and (P (Y ) , l ). Proof. We have to prove the properties from the Definition 7 of an antitone Galois connection . First, we show that if A1 , A2 , A X then: (i) A1 A2 implies l (A2 ) l l (A1 ) ( l is antitone) and (ii) A l (l (A)) (l l is extensive) The proof of (i) is straightforward. If A = by the definition of the derivation operators we have l (l (A)) = R (x) x X| R (x) R (x) , which includes A. In the case A = the relation (ii) is obviously fullfilled. Next if B1 , B2 , B X, we prove that: (iii) B1 l B2 implies l (B2 ) l (B1 ) ( l is antitone) and (iv) B l l (l (B)) (l l is extensive). If x l (B2 ) , it follows that B2 l R (x) which together with B1 l B2 implies that x l (B1 ). We now show that the operator l l is extensive. If l (B) = , for all x l (B) , we have B R (x) and consequently B xl (B) R (x) = l (l (B)) , inclusion which proves the conclusion. In the case l (B) = , l (l (B)) = Y and it is obvious that B l Y. Remark 12. The pair (l , l ) form an antitone Galois connection between the spaces (P (X) , ) and (P (Y ) , l ) if and only if, for all A P (X) and B P (Y ) , A l (B) l (A) l B (this is a classical result from the theory of Galois connections). Proposition 13. The maps l l , l l are closure operators on the powersets (P (X) , ) and (P (Y ) , l ) , respectively. Proof. Due to symmetry we prove only the first property. Because l and l are antitones it results that l l is isotone. The extensivity of the operator l l was proved in theorem 11 (property (ii) ). We prove now the equality (1) l (A) =l l (l (l (A))) , A P (X) , which because of the monotonicity of the operator l implies the idempotency of l l . By A l (l (A)) and theorem 11 (property (i)) it follows that l (A) l l (l (l (A))) . To prove the reverse inequality we first notice that l (l (l (A))) = R (x) . xl (l (A)) For all x l (l (A)) , using the definition of the operator l we have l (A) l R (x) or, equivalently, l (A) R (x) . Thus it results the relation l (A) R (x) xl (l (A)) which proves that l (A) l l (l (l (A))) . We define the notion of lower concept. Definition 14. Let (X, (Y, ) , R) be an attribute ordered formal context. A pair (A, B) P (X) × P (Y ) is called a lower formal concept in (X, (Y, ) , R) if l (A) =l B and l (B) = A. We denote the set of lower formal concepts by Cl (X, (Y, ) , R). We define a partial order l between lower concepts; we say that the concept (A1 , B1 ) is less (more specific) than the concept (A2 , B2 ) or that (A2 , B2 ) is greater (more general) than (A1 , B1 )) if A1 A2 (equivalently B2 l B1 ). Remark 15. A pair (A, B) is a lower formal concept if and only if A is a fixed point of the closure operator l l . As we show next, the fundamental theorem of the FCA still holds for the set of lower formal concepts. Theorem 16. Let (X, (Y, ) , R) be an attribute ordered formal context. The set Cl (X, (Y ) , R) endowed with the relation l is a complete lattice, which we call lower concept lattice or lower Galois lattice. This lattice is denoted by Cl (X, (Y, ) , R). Proof. We prove that for all (Ai )iI X, I an index set, the relation (2) l ( iI Ai ) =l inf l l (Ai ) or, equivalently, { R (x) |x iI Ai } = iI ( i R (x)) holds. An attribute y belongs to the set { R (x) |x iI Ai } if and only if, for all i I, for all x Ai , y R (x) , which in turn is true if and only if y iI ( i R (x)). Applying now the definitions of the derivation operators we obtain: l (sup l {Bi |i I}) = {x X| sup l {Bi |i I} l R (x)} = iI {x X|Bi l R (x)} = iI l (Bi ) , thus let us notice the relation (3) l (sup l {Bi |i I}) = iI l (Bi ) . We prove that Cl (X, (Y ) , R) is a complete lattice. If {(Ai , Bi )|i I} is a family of lower formal concepts we define the operators and by: (Ai , Bi ) = ( Ai , l (l (sup l {Bi |i I})), Ai )), inf l {Bi |i I}). (Ai , Bi ) = (l (l ( We first show that iI (Ai , Bi ) is a lower formal concept. Using the definition of a lower formal concept and the relation (3) we have l ( iI Ai ) =l l ( iI l (Bi )) =l l (l (sup l {Bi |i I})) . By Proposition 13 (the relation similar to (1)) and equality (3) it follows that: l (l (l (sup l {Bi |i I}))) =l l (sup l {Bi |i I}) = iI l (Bi ) = iI Ai . To prove that (1) and (2): l (l (l ( iI (Ai , Bi ) is a lower formal concept we use the relations Ai ) =l inf l l (Ai ) Ai ))) =l l ( iI =l inf {Bi |i I}. iI We have also l (inf l {Bi |i I}) = l (inf l {l (Ai )|i I}) = l (l ( iI Ai )). By the definition of the partial order l in Cl (X, (Y ) , R) , it is clear that the lower concept iI (Ai , Bi ) is the infimum of the family {(Ai , Bi )|i I} , while iI (Ai , Bi ) is the supremum of the same family. Remark 17. The lower concept lattice generalizes the similar classical notion; for if the order relation on Y is equal with Y (the diagonal of Y ) then the two notions are the same. 4. Upper formal concepts We consider an attribute ordered formal context (X, (Y, ) , R) and redefine the derivation operators from the classical case by changing the space (P (Y ) , ) with the powerdomain (P (Y ) , u ) . We define another type of derivation operators: Definition 18. Let (X, (Y, ) , R) be an attribute ordered formal context. The operators u : (P (X) , ) (P (Y ) , u ) , u : (P (Y ) , u ) (P (X) , ) defined by: u (A) = inf {R (x) |x A} , A X, A = and u () = , u (B) = {x X|B u R (x)} , B P (Y ) are called the upper derivation operators. We denoted by inf obtained in P (Y ) considering the preorder u . u the infimum Theorem 19. The pair (u , u ) forms an antitone Galois connection between the powersets (P (X) , ) and (P (Y ) , u ). Proof. It is obvious that the derivations operators u and u are antitone operators. To prove A u (u (A)) , A P (X) , A = , we use the equivalence (ii) from proposition 6 and we have: u (u (A)) = u ( { R (x) |x A}) = x X| ( R (x)) R (x) . It is clear that this set includes A. The proof is straightforward in the case A = . We show now that the operator u u is extensive. Thus, for all x u (B) , we have B R (x) and consequently B xu (B) R (x) = u (u (B)) , inclusion which implies B u u (u (B)). Proposition 20. The maps u u , u u are closure operators on the powersets (P (X) , ) and (P (Y ) , u ) , respectively. Proof. Because u and u are antitones it results that u u and u u are isotone. The extensivity of the operators u u and u u is been proven in theorem 19. We prove now the equality (4) u (A) =u u (u (u (A))) , A P (X) , which together with u antitone imply the idempotency of u u . By A u (u (A)) and u antitone it follows that u (A) u u (u (u (A))) . To prove the reverse inequality we notice first that u (u (u (A))) = xu (u (A)) R (x) . For all x u (u (A)), using the definition of the operator u we have u (A)u R(x) or, equivalently, u (A)R(x). Thus it results the inclusion u (A) xu (u (A)) R (x) which proves u (A) u u (u (u (A))) . The idempotency of u u can be prove by a similar reasoning. We now introduce the notion of upper concept. Definition 21. Let (X, (Y, ) , R) be an attribute ordered formal context. A pair (A, B) P (X) × P (Y ) is called an upper formal concept in (X, (Y, ) , R) if u (A) =u B and u (B) = A. We denote the set of upper formal concepts by Cu (X, (Y, ) , R). We define a partial order u between the upper concepts: we say that the concept (A1 , B1 ) is less (more specific) than the concept (A2 , B2 ) or that (A2 , B2 ) is greater (more general) than (A1 , B1 )) if A1 A2 (equivalently B2 u B1 ). Theorem 22. Let (X, (Y, ) , R) be an attribute ordered formal context. The set Cu (X, (Y ) , R) endowed with the relation u is a complete lattice, which we call upper concept lattice or upper Galois lattice. This lattice is denoted by C u (X, Y, I). Proof. We prove that for all (Ai )iI X, I an index set, the relation (5) u ( iI Ai ) =u inf u u (Ai ), or equivalently { R (x) |x iI Ai } = iI ( i R (x)) holds. An attribute y belongs to the set { R (x) |x iI Ai } if and only if, it exists x iI Ai , such that y R (x) , which in turn is true if and only if y iI ( i R (x)). In a similar way one can prove that (6) u (sup(u {Bi |i I}) = iI u (Bi ) . We prove now that Cu (X, (Y ), R) is a complete lattice. If {(Ai , Bi )|i I} is a family of upper formal concepts we define the operators and by: (Ai , Bi ) = ( Ai , u (u (supu {Bi |i I})), Ai ), inf u {Bi |i (Ai , Bi ) = (u (u ( I}). By a similar reasoning as in Theorem 16, using Proposition 20 (the relation (4)) and equalities (5) and (6) one can prove that iI (Ai , Bi ) and iI (Ai , Bi ) are upper formal concepts. Taking into account that the extent of iI (Ai , Bi ) is iI Ai it follows that the element iI (Ai , Bi ) Cu (X, (Y ) , R) is the infimum of the family {(Ai , Bi )|i I}. Analogously, it results that iI (Ai , Bi ) is the supremum of the same family. The lower and the upper concept lattices we have already defined and the classical concept lattice offer different views in order to analyze the relationships between objects, according to their properties, as the following emple shows: 5. An emple We consider the set of objects X = {d1 , d2 , d3 , d4 }, where d1 , d2 , d3 , d4 are documents (web pages, for emple) and Y = {a1 , a2 , a3 , a4 , a5 , a6 , a7 } a set of terms. The set of attributes Y is endowed with the partial order given by = Y {(a1 , a2 ) , (a1 , a3 ) , (a4 , a5 ) , (a7 , a5 ) , (a7 , a6 )}. The incidence relation in the table below describes how the attributes from Y are assigned to the web pages from X. d1 d2 d3 d4 a1 × a2 × × × a3 × × a4 a5 × × a6 × × a7 × × × × There are many algorithms to find all the concepts of a concept lattice in the classical setting such are the AddIntent algorithm [10] or the Lindig's algorithm LATTICE [9]. In our case to build the lower (upper) concept lattice we adjust the Ganter's NextClosure algorithm [6], which generates all closures of a closure operator, to our requirements. We find all extents for the given context as fixed points of the closure operators l l , respectively u u . The concepts for the classical concept lattice are: c1 = ({d1 , d2 , d3 , d4 }, ) (the top concept) c2 = ({d1 , d2 , d3 }, {a2 }) c3 = ({d2 , d4 }, {a2 , a3 }) c4 = ({d1 , d2 }, {a2 , a5 , a7 }) , c5 = ({d1 , d4 }, {a1 }) c6 = ({d1 , d3 }, {a2 , a6 }) , c7 = ({d3 , d4 }, {a4 }) c8 = ({d2 }, {a2 , a3 , a5 , a7 }) , c9 = ({d1 }, {a1 , a2 , a5 , a6 , a7 }) c10 = ({d4 }, {a1 , a4 }) , c11 = ({d3 }, {a2 , a4 , a6 }) c12 = (, {a1 , a2 , a3 , a4 , a5 , a6 , a7 }) (the bottom concept) The concepts for the lower concept lattice are: c1 = ({d1 , d2 , d3 , d4 }, {a2 , a5 }) (the top concept) c2 = ({d1 , d2 , d4 }, {a2 , a3 , a5 }) , c3 = ({d1 , d2 , d3 }, {a2 , a5 , a6 }) c4 = ({d3 , d4 }, {a2 , a4 , a5 }) , c5 = ({d1 , d4 }, {a1 , a2 , a3 , a5 }) c6 = ({d1 , d2 }, {a2 , a3 , a5 , a7 }) , c7 = ({d1 }, {a1 , a2 , a3 , a5 , a6 , a7 }, ) c8 = ({d4 }, {a1 , a2 , a3 , a4 , a5 }) , c9 = ({d3 }, {a2 , a4 , a5 , a6 }) , c10 = (, {a1 , a2 , a3 , a4 , a5 , a6 , a7 }) (the bottom concept). If (A, B) is a concept, then we interpret B as a set of terms appearing in a query, while A represents the documents retrieved. The notions of classical concept lattice and lower (upper) concept lattice are different. Thus the concepts associated to the document d4 are different in the two settings: (d4 ) = ({d4 }, {a1 , a4 }) , while l (d4 ) = ({d4 }, {a1 , a2 , a3 , a4 , a5 }) . One can notice also that in the lower case, due to the new type of the derivation operators, the objects may be rearranged in different clusters. Consequently, the lower lattice may contain new concepts as c2 or miss some ones from the classical lattice like are c3 , c6, c8 . The lower lattice provides also another type of hierarchy between its concepts. If we use this lattice we obtain different approximations for the query submitted as in the classical case, thus improving the search process in the database (which is represented by the formal context). For emple, if we address to the system the query q = {a2 , a6 }, in the classical case we have q = intent (c6 ) and obtain as response the set of documents {d1 , d3 }, while in the lower concept lattice we have to expand q to the query q = {a2 , a5 , a6 } with the response {d1 , d2 , d3 }. We emphasize also that unlike the classical setting in the case of lower concept lattices bigger classes of objects are described by more specific attributes. The extent of the lower concept c3 , {d1 , d2 , d3 }, is described by its intent {a2 , a5 , a6 }, which contains more specific properties than {a2 , a3 , a5 , a7 } the intent of the smaller concept c6 . In fact, we have c6 l c3 which is equivalent with {a2 , a5 , a6 } l {a2 , a3 , a5 , a7 }. By changing the attribute a6 with the more general term a7 , one obtains the better approximation {a2 , a3 , a5 , a7 }. Thus, the Hoare hierarchy corresponds to a generative sequence of properties. On the contrary, in the case of upper concepts one obtains better approximations by replacing attributes with more specific terms. 6. Conclusion and future work In this paper we have proposed two new types of conceptual structures, which provide new methods for data analysis and knowledge representation. Based on two new types of derivation operators and on notions of domain theory we have defined two Galois connections and introduced the notions of lower and, respectively, upper formal concept. These new structures have analogous properties to those of the classical concept lattices. Thus we have proved that the fundamental theorem of the Formal Concept Analysis still holds in this new setting. However, due to the new type of formal concepts, these structures provide a different way to organize, to find and to vizualize the information. Using the conceptual hierarchy derived from the Hoare and Smyth preorder we get additional data concerning the connections between the objects when dealing with their properties. It is well-known [6] that, given a concept lattice, one can read off from the hierarchical structure some data dependencies between attributes. Thus, it will be interesting to study how these implications between the properties of the objects change when considering the set of intents concepts ordered with the Hoare (Smyth) preorder. The complexity of the lower (upper) concept lattices depends on the attribute ordered formal context and on the order relation between the attributes. To investigate the dependency of the size of the concept lattice on small/medium contexts to these parameters could be another research direction. There are many methods [1], [5] to reduce the size of the concept lattice in different settings. We want also to study how these techniques can be applied to our work. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Annals of the Alexandru Ioan Cuza University - Mathematics de Gruyter

Lower and Upper Concept Lattices

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de Gruyter
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1221-8421
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1221-8421
DOI
10.2478/aicu-2013-0024
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Abstract

Formal Concept Analysis is a mathematical theory of data analysis using formal contexts and concept lattices. In this paper, two new types of concept lattices are introduced by using notions from domain theory (in particular, Hoare and Smyth powerdomains). Based on a Galois connection, we prove the fundamental theorem of the Formal Concept Analysis, as well as other properties of lower and upper formal concepts. In this way, we provide new models to represent and retrieve the information in data and knowledge systems. Mathematics Subject Classification 2010: 03G10, 06A15, 68P20. Key words: concept lattice, Galois connection, powerdomain, data analysis. 1. Introduction Formal Concept Analysis (FCA) is a mathematical theory which analyze the relationships between a set of objects and the set of their properties (attributes) and offers a way to organize, to build and to represent conceptual structures. As a result of investigating and processing the given information the FCA methods build a collection of concepts which are the natural clusters of the objects, according to their attributes. This set of formal concepts may be partially ordered, thus a system of sub- and superconcepts, a conceptual hierarchy, can be obtained. Methods to discover and to analyze dependencies between data and to vizualize these informations are also provided by this approach. All these features make FCA a powerful tool which has been succesfully applied in a wide range of application fields. In the last decade, FCA has been providing a support structure for information retrieval, web search, data mining, gene expression data analysis, latent semantic indexing. Formal concept analysis was proposed by Wille [12] as a mathematical method of data analysis. This approach takes a binary relation between a set of objects and a set of attributes, which are properties of the objects and creates a space of "concepts". Giving to these sets a concrete meaning, one can get different ways to model the information; for emple, one can consider the objects as documents (for emple web pages), while the attributes are terms. A natural idea which was first developed by Burusco and FuentesGonzales [4] was to consider fuzzy contexts in which one assigns to every object a truth degree to which the object has an attribute. They proved that the set of fuzzy concepts is a complete lattice and give a constructive method to build this lattice. However it has been proven that some of the properties from the classical formal concept theory do not hold in this new setting. Pollandt [11] and Belohlavek [2] independently introduced another type of formal concept based on a context with values in a complete residuated lattice, which proved to be a feasibile approach. Based on a Galois connection they proved that all important results from the classical theory of Wille still hold in their setting. In order to reduce the size of the concept lattice which corresponds to large contexts Belohlavek et al. [3] proposed a new approach which generalizes both the fuzzy concept lattice and the lattice of crisply generated fuzzy concepts. Thus, to reduce the complexity of the concept lattice they used as parameters two unary functions defined on the residuated lattice called hedges. Another method to choose the significant data from a very large context was introduced by Zhang et al. [13], which defined three kinds of variable threshold concept lattices. In [8] Krajci constructed the so-called generalized concept lattice. He defined the notion of concept by considering a three-sorted structure of truth degrees which does not include the implication operators. Krajci showed that every concept lattice with hedges is isomorphic to some generalized concept lattice. Another interesting approach was proposed by Georgescu and Popescu [7] which developed the theory of rough concept lattices in a fuzzy setting. In all the articles we have previously mentioned the notion of formal concept was defined as a fixed point of a Galois connection between two partially ordered spaces of sets. Both in classical and fuzzy formal contexts these spaces of sets were endowed with the inclusion order. In our article we define two new types of derivation operators which form a Galois connection between the powerset of objects and the powerset of attributes but endowed with two different types of orderings. On the powerset of attributes we consider the Hoare and, respectively, the Smyth preorders, thus getting the Hoare (lower) and, respectively, the Smyth (upper) powerdomains. Based on the Galois connection and on notions from domain theory, we define two new types of formal concepts, namely the lower and the upper formal concepts and study the structure of the set of concepts. In this new setting we prove the fundamental theorem of the FCA, which states that the set of concepts is a complete lattice, as well as other properties of lower and upper formal concepts. These new types of conceptual structures provide new models to cluster, analyze and represent the information in databases. It is known that concept lattices are mostly used for IR applications, to improve the search strategies like query refinement, ranking or document classification. In these tasks, a query submitted by a user is considered as a concept intent, while the documents retrieved by the system are assimilated with its extent. In our approach the hierarchy in the concept lattices is derived from the Hoare and, respectively, the Smyth preorders and, consequently, it is different from the conceptual structures noticed above. This results in different approximations for a query (concept intent) addressed to a search system as in the classical setting. The paper is organized as it follows. In Section 2 we briefly overview the classical Formal Concept Analysis results and some basic notions about the Hoare and Smyth preorders. Then we introduce the notions of lower and upper formal concept and study their properties in Sections 3 and, respectively 4. An emple for the theoretical model is presented in Section 5. 2. Preliminaries 2.1. Basics of formal concept analysis Definition 1. A formal context is a triple (X, Y, R) with X,Y being abstract sets and R X × Y a relation between X and Y . The elements of X are called objects, those of Y attributes and R the incidence relation of the context (X, Y, R). If xRy we say that "the object x has the attribute y" ( or "y is an attribute of object x"). We define now the derivation operators: Definition 2. Let (X, Y, R) be a formal context. The applications : (P (X) , ) (P (Y ) , ) and : (P (Y ) , ) (P (X) , ) defined by (A) = {y Y | x A, xRy} , A = , () = Y (B) = {x X | y B, xRy} , B = , () = X are called the derivation operators. The set (A) Y, denotes the set of attributes common to all objects in A, while (B) denotes the set of objects which have every attribute in B. It is known that the derivation operators , form an antitone Galois connection between the posets (P (X) , ) and (P (Y ) , ). Definition 3. Let (X, Y, R) be a formal context. A pair (A, B) P (X) × P (Y ) is said to be a formal concept of (X, Y, R) if (A) = B and (B) = A. The sets A, B are called the extent and, respectively, the intent of the formal concept (A, B).The set of all formal concepts is denoted by C (X, Y, R) . If (A, B) is a formal concept, then the sets A, B are maximal according to the properties stated in the definitions. Every formal concept can be obtained as a pair ( ((A)) , (A)), A P (X) or as a pair ( (B) , ( (B))), B P (Y ) . On the set C (X, Y, R) a partial order is been defined; thus the concept (A1 , B1 ) is said to be less than the concept (A2 , B2 ) if A1 A2 . The fundamental theorem of FCA states that the set of formal concepts is a complete lattice: Theorem 4. The set C (X, Y, R) endowed with the relation is a complete lattice which is called concept lattice or Galois lattice. This lattice is denoted by C(X, Y, R). The proof of this result can be found in [6]. 2.2. Hoare and Smyth preorders In this subsection, we consider a partially ordered set (X, ) and we denote by Pf (X) the collection of finite subsets of X. If A X we denote by A, (respectively by A), the lower (upper) closure of the set A i.e. the sets A = {x X|a A, x a}, A = {x X|a A, a x}. We remember the Hoare and Smyth preorderings on Pf (X), which will be important in our study. Definition 5. For all A, B Pf (X) we define the relations l and u on the powerset Pf (X) by A l B if a A, b B, such that a b, and A u B if b B, a A such that a b. The preorders (Pf (X) , l ) and (Pf (X) , u ) are called the Hoare (or lower) and, respectively, the Smyth (or upper) powerdomains of (X, ). It is wellknown that powerdomains have been used for defining a semantics for the programming languages with non-determinism, but in our study we use them as a mathematical support for approximation in partially ordered sets. We may write B l A rather than A l B and =l if A l B and B l A. Similar denotations are used for the upper preorder. Proposition 6. Let (X, ) be a partially ordered set. For all A, B Pf (X) the next equivalences hold: i.A ii.A lB uB if and only if A B if and only if A B Definition 7. Let f : (X, ) (Y, ) and g : (Y, ) (X, ) be two maps between two preordered sets (X, ) , (Y, ). The pair (f, g) is called an antitone Galois connection if 1. for all x1 , x2 X, x1 x2 , f (x1 ) f (x2 ) (f is antitone); 2. for all y1 , y2 Y , y1 y2 , g (y1 ) g (y2 ) (g is antitone); 3. for all x X, x g (f (x)) ( g f is extensive); 4. for all y Y, y f (g (y)) ( f g is extensive). The two maps are called dually adjoint to each other. The above definition is slightly different than the usual one, because (X, ) , (Y, ) are only preordered sets. We notice that the results we are interested to are still true in this case. 3. Lower formal concepts In what follows we consider X a set of objects and Y a finite set of properties (attributes) of objects. In Information Retrieval and Data Mining applications one often consider X to be a set of documents (web pages, for emple), while Y is a hierarchy of terms describing the documents in the form of a thesaurus. Thus it is natural to endow the set of attributes Y with a partial order which can be interpreted as a specialization/generalization relation between the elements of Y. The relation y1 y2 means that the attribute y1 specializes the attribute y2 or that y2 generalizes y1 . Definition 8. An attribute ordered formal context is a triple (X, (Y, ), R) with R X × Y being a relation between X and Y . We call R the incidence relation of the context (X, (Y, ) , R). If xRy we say that "the object x has the attribute y" ( or "y is an attribute of object x"). If x X, we denote the set {y Y | xRy} by R (x) . Using a different order on the powerset of attributes we introduce a new type of formal concept and study the properties of the corresponding concept lattice. First, we define two new derivation operators type: Definition 9. Let (X, (Y, ) , R) be an attribute ordered formal context. The operators l : (P (X) , ) (P (Y ) , l ) and l : (P (Y ) , l ) (P (X) , ) defined by: l (A) = inf {R (x) |x A} , A X, A = and l () = Y, l the l (B) = {x X|B l R (x)} , B P (Y ) , are called lower derivation operators. We denoted by inf obtained in P (Y ) considering the preorder l . infimum Remark 10. By proposition 6 (i) we have l (A) = R (x) , hence y l (A) means that for all x A, x has an attribute y such that y y. An object x is in l (B) if for every y B, x has an attribute y such that y y. Theorem 11. The pair (l , l ) form an antitone Galois connection between the powersets (P (X) , ) and (P (Y ) , l ). Proof. We have to prove the properties from the Definition 7 of an antitone Galois connection . First, we show that if A1 , A2 , A X then: (i) A1 A2 implies l (A2 ) l l (A1 ) ( l is antitone) and (ii) A l (l (A)) (l l is extensive) The proof of (i) is straightforward. If A = by the definition of the derivation operators we have l (l (A)) = R (x) x X| R (x) R (x) , which includes A. In the case A = the relation (ii) is obviously fullfilled. Next if B1 , B2 , B X, we prove that: (iii) B1 l B2 implies l (B2 ) l (B1 ) ( l is antitone) and (iv) B l l (l (B)) (l l is extensive). If x l (B2 ) , it follows that B2 l R (x) which together with B1 l B2 implies that x l (B1 ). We now show that the operator l l is extensive. If l (B) = , for all x l (B) , we have B R (x) and consequently B xl (B) R (x) = l (l (B)) , inclusion which proves the conclusion. In the case l (B) = , l (l (B)) = Y and it is obvious that B l Y. Remark 12. The pair (l , l ) form an antitone Galois connection between the spaces (P (X) , ) and (P (Y ) , l ) if and only if, for all A P (X) and B P (Y ) , A l (B) l (A) l B (this is a classical result from the theory of Galois connections). Proposition 13. The maps l l , l l are closure operators on the powersets (P (X) , ) and (P (Y ) , l ) , respectively. Proof. Due to symmetry we prove only the first property. Because l and l are antitones it results that l l is isotone. The extensivity of the operator l l was proved in theorem 11 (property (ii) ). We prove now the equality (1) l (A) =l l (l (l (A))) , A P (X) , which because of the monotonicity of the operator l implies the idempotency of l l . By A l (l (A)) and theorem 11 (property (i)) it follows that l (A) l l (l (l (A))) . To prove the reverse inequality we first notice that l (l (l (A))) = R (x) . xl (l (A)) For all x l (l (A)) , using the definition of the operator l we have l (A) l R (x) or, equivalently, l (A) R (x) . Thus it results the relation l (A) R (x) xl (l (A)) which proves that l (A) l l (l (l (A))) . We define the notion of lower concept. Definition 14. Let (X, (Y, ) , R) be an attribute ordered formal context. A pair (A, B) P (X) × P (Y ) is called a lower formal concept in (X, (Y, ) , R) if l (A) =l B and l (B) = A. We denote the set of lower formal concepts by Cl (X, (Y, ) , R). We define a partial order l between lower concepts; we say that the concept (A1 , B1 ) is less (more specific) than the concept (A2 , B2 ) or that (A2 , B2 ) is greater (more general) than (A1 , B1 )) if A1 A2 (equivalently B2 l B1 ). Remark 15. A pair (A, B) is a lower formal concept if and only if A is a fixed point of the closure operator l l . As we show next, the fundamental theorem of the FCA still holds for the set of lower formal concepts. Theorem 16. Let (X, (Y, ) , R) be an attribute ordered formal context. The set Cl (X, (Y ) , R) endowed with the relation l is a complete lattice, which we call lower concept lattice or lower Galois lattice. This lattice is denoted by Cl (X, (Y, ) , R). Proof. We prove that for all (Ai )iI X, I an index set, the relation (2) l ( iI Ai ) =l inf l l (Ai ) or, equivalently, { R (x) |x iI Ai } = iI ( i R (x)) holds. An attribute y belongs to the set { R (x) |x iI Ai } if and only if, for all i I, for all x Ai , y R (x) , which in turn is true if and only if y iI ( i R (x)). Applying now the definitions of the derivation operators we obtain: l (sup l {Bi |i I}) = {x X| sup l {Bi |i I} l R (x)} = iI {x X|Bi l R (x)} = iI l (Bi ) , thus let us notice the relation (3) l (sup l {Bi |i I}) = iI l (Bi ) . We prove that Cl (X, (Y ) , R) is a complete lattice. If {(Ai , Bi )|i I} is a family of lower formal concepts we define the operators and by: (Ai , Bi ) = ( Ai , l (l (sup l {Bi |i I})), Ai )), inf l {Bi |i I}). (Ai , Bi ) = (l (l ( We first show that iI (Ai , Bi ) is a lower formal concept. Using the definition of a lower formal concept and the relation (3) we have l ( iI Ai ) =l l ( iI l (Bi )) =l l (l (sup l {Bi |i I})) . By Proposition 13 (the relation similar to (1)) and equality (3) it follows that: l (l (l (sup l {Bi |i I}))) =l l (sup l {Bi |i I}) = iI l (Bi ) = iI Ai . To prove that (1) and (2): l (l (l ( iI (Ai , Bi ) is a lower formal concept we use the relations Ai ) =l inf l l (Ai ) Ai ))) =l l ( iI =l inf {Bi |i I}. iI We have also l (inf l {Bi |i I}) = l (inf l {l (Ai )|i I}) = l (l ( iI Ai )). By the definition of the partial order l in Cl (X, (Y ) , R) , it is clear that the lower concept iI (Ai , Bi ) is the infimum of the family {(Ai , Bi )|i I} , while iI (Ai , Bi ) is the supremum of the same family. Remark 17. The lower concept lattice generalizes the similar classical notion; for if the order relation on Y is equal with Y (the diagonal of Y ) then the two notions are the same. 4. Upper formal concepts We consider an attribute ordered formal context (X, (Y, ) , R) and redefine the derivation operators from the classical case by changing the space (P (Y ) , ) with the powerdomain (P (Y ) , u ) . We define another type of derivation operators: Definition 18. Let (X, (Y, ) , R) be an attribute ordered formal context. The operators u : (P (X) , ) (P (Y ) , u ) , u : (P (Y ) , u ) (P (X) , ) defined by: u (A) = inf {R (x) |x A} , A X, A = and u () = , u (B) = {x X|B u R (x)} , B P (Y ) are called the upper derivation operators. We denoted by inf obtained in P (Y ) considering the preorder u . u the infimum Theorem 19. The pair (u , u ) forms an antitone Galois connection between the powersets (P (X) , ) and (P (Y ) , u ). Proof. It is obvious that the derivations operators u and u are antitone operators. To prove A u (u (A)) , A P (X) , A = , we use the equivalence (ii) from proposition 6 and we have: u (u (A)) = u ( { R (x) |x A}) = x X| ( R (x)) R (x) . It is clear that this set includes A. The proof is straightforward in the case A = . We show now that the operator u u is extensive. Thus, for all x u (B) , we have B R (x) and consequently B xu (B) R (x) = u (u (B)) , inclusion which implies B u u (u (B)). Proposition 20. The maps u u , u u are closure operators on the powersets (P (X) , ) and (P (Y ) , u ) , respectively. Proof. Because u and u are antitones it results that u u and u u are isotone. The extensivity of the operators u u and u u is been proven in theorem 19. We prove now the equality (4) u (A) =u u (u (u (A))) , A P (X) , which together with u antitone imply the idempotency of u u . By A u (u (A)) and u antitone it follows that u (A) u u (u (u (A))) . To prove the reverse inequality we notice first that u (u (u (A))) = xu (u (A)) R (x) . For all x u (u (A)), using the definition of the operator u we have u (A)u R(x) or, equivalently, u (A)R(x). Thus it results the inclusion u (A) xu (u (A)) R (x) which proves u (A) u u (u (u (A))) . The idempotency of u u can be prove by a similar reasoning. We now introduce the notion of upper concept. Definition 21. Let (X, (Y, ) , R) be an attribute ordered formal context. A pair (A, B) P (X) × P (Y ) is called an upper formal concept in (X, (Y, ) , R) if u (A) =u B and u (B) = A. We denote the set of upper formal concepts by Cu (X, (Y, ) , R). We define a partial order u between the upper concepts: we say that the concept (A1 , B1 ) is less (more specific) than the concept (A2 , B2 ) or that (A2 , B2 ) is greater (more general) than (A1 , B1 )) if A1 A2 (equivalently B2 u B1 ). Theorem 22. Let (X, (Y, ) , R) be an attribute ordered formal context. The set Cu (X, (Y ) , R) endowed with the relation u is a complete lattice, which we call upper concept lattice or upper Galois lattice. This lattice is denoted by C u (X, Y, I). Proof. We prove that for all (Ai )iI X, I an index set, the relation (5) u ( iI Ai ) =u inf u u (Ai ), or equivalently { R (x) |x iI Ai } = iI ( i R (x)) holds. An attribute y belongs to the set { R (x) |x iI Ai } if and only if, it exists x iI Ai , such that y R (x) , which in turn is true if and only if y iI ( i R (x)). In a similar way one can prove that (6) u (sup(u {Bi |i I}) = iI u (Bi ) . We prove now that Cu (X, (Y ), R) is a complete lattice. If {(Ai , Bi )|i I} is a family of upper formal concepts we define the operators and by: (Ai , Bi ) = ( Ai , u (u (supu {Bi |i I})), Ai ), inf u {Bi |i (Ai , Bi ) = (u (u ( I}). By a similar reasoning as in Theorem 16, using Proposition 20 (the relation (4)) and equalities (5) and (6) one can prove that iI (Ai , Bi ) and iI (Ai , Bi ) are upper formal concepts. Taking into account that the extent of iI (Ai , Bi ) is iI Ai it follows that the element iI (Ai , Bi ) Cu (X, (Y ) , R) is the infimum of the family {(Ai , Bi )|i I}. Analogously, it results that iI (Ai , Bi ) is the supremum of the same family. The lower and the upper concept lattices we have already defined and the classical concept lattice offer different views in order to analyze the relationships between objects, according to their properties, as the following emple shows: 5. An emple We consider the set of objects X = {d1 , d2 , d3 , d4 }, where d1 , d2 , d3 , d4 are documents (web pages, for emple) and Y = {a1 , a2 , a3 , a4 , a5 , a6 , a7 } a set of terms. The set of attributes Y is endowed with the partial order given by = Y {(a1 , a2 ) , (a1 , a3 ) , (a4 , a5 ) , (a7 , a5 ) , (a7 , a6 )}. The incidence relation in the table below describes how the attributes from Y are assigned to the web pages from X. d1 d2 d3 d4 a1 × a2 × × × a3 × × a4 a5 × × a6 × × a7 × × × × There are many algorithms to find all the concepts of a concept lattice in the classical setting such are the AddIntent algorithm [10] or the Lindig's algorithm LATTICE [9]. In our case to build the lower (upper) concept lattice we adjust the Ganter's NextClosure algorithm [6], which generates all closures of a closure operator, to our requirements. We find all extents for the given context as fixed points of the closure operators l l , respectively u u . The concepts for the classical concept lattice are: c1 = ({d1 , d2 , d3 , d4 }, ) (the top concept) c2 = ({d1 , d2 , d3 }, {a2 }) c3 = ({d2 , d4 }, {a2 , a3 }) c4 = ({d1 , d2 }, {a2 , a5 , a7 }) , c5 = ({d1 , d4 }, {a1 }) c6 = ({d1 , d3 }, {a2 , a6 }) , c7 = ({d3 , d4 }, {a4 }) c8 = ({d2 }, {a2 , a3 , a5 , a7 }) , c9 = ({d1 }, {a1 , a2 , a5 , a6 , a7 }) c10 = ({d4 }, {a1 , a4 }) , c11 = ({d3 }, {a2 , a4 , a6 }) c12 = (, {a1 , a2 , a3 , a4 , a5 , a6 , a7 }) (the bottom concept) The concepts for the lower concept lattice are: c1 = ({d1 , d2 , d3 , d4 }, {a2 , a5 }) (the top concept) c2 = ({d1 , d2 , d4 }, {a2 , a3 , a5 }) , c3 = ({d1 , d2 , d3 }, {a2 , a5 , a6 }) c4 = ({d3 , d4 }, {a2 , a4 , a5 }) , c5 = ({d1 , d4 }, {a1 , a2 , a3 , a5 }) c6 = ({d1 , d2 }, {a2 , a3 , a5 , a7 }) , c7 = ({d1 }, {a1 , a2 , a3 , a5 , a6 , a7 }, ) c8 = ({d4 }, {a1 , a2 , a3 , a4 , a5 }) , c9 = ({d3 }, {a2 , a4 , a5 , a6 }) , c10 = (, {a1 , a2 , a3 , a4 , a5 , a6 , a7 }) (the bottom concept). If (A, B) is a concept, then we interpret B as a set of terms appearing in a query, while A represents the documents retrieved. The notions of classical concept lattice and lower (upper) concept lattice are different. Thus the concepts associated to the document d4 are different in the two settings: (d4 ) = ({d4 }, {a1 , a4 }) , while l (d4 ) = ({d4 }, {a1 , a2 , a3 , a4 , a5 }) . One can notice also that in the lower case, due to the new type of the derivation operators, the objects may be rearranged in different clusters. Consequently, the lower lattice may contain new concepts as c2 or miss some ones from the classical lattice like are c3 , c6, c8 . The lower lattice provides also another type of hierarchy between its concepts. If we use this lattice we obtain different approximations for the query submitted as in the classical case, thus improving the search process in the database (which is represented by the formal context). For emple, if we address to the system the query q = {a2 , a6 }, in the classical case we have q = intent (c6 ) and obtain as response the set of documents {d1 , d3 }, while in the lower concept lattice we have to expand q to the query q = {a2 , a5 , a6 } with the response {d1 , d2 , d3 }. We emphasize also that unlike the classical setting in the case of lower concept lattices bigger classes of objects are described by more specific attributes. The extent of the lower concept c3 , {d1 , d2 , d3 }, is described by its intent {a2 , a5 , a6 }, which contains more specific properties than {a2 , a3 , a5 , a7 } the intent of the smaller concept c6 . In fact, we have c6 l c3 which is equivalent with {a2 , a5 , a6 } l {a2 , a3 , a5 , a7 }. By changing the attribute a6 with the more general term a7 , one obtains the better approximation {a2 , a3 , a5 , a7 }. Thus, the Hoare hierarchy corresponds to a generative sequence of properties. On the contrary, in the case of upper concepts one obtains better approximations by replacing attributes with more specific terms. 6. Conclusion and future work In this paper we have proposed two new types of conceptual structures, which provide new methods for data analysis and knowledge representation. Based on two new types of derivation operators and on notions of domain theory we have defined two Galois connections and introduced the notions of lower and, respectively, upper formal concept. These new structures have analogous properties to those of the classical concept lattices. Thus we have proved that the fundamental theorem of the Formal Concept Analysis still holds in this new setting. However, due to the new type of formal concepts, these structures provide a different way to organize, to find and to vizualize the information. Using the conceptual hierarchy derived from the Hoare and Smyth preorder we get additional data concerning the connections between the objects when dealing with their properties. It is well-known [6] that, given a concept lattice, one can read off from the hierarchical structure some data dependencies between attributes. Thus, it will be interesting to study how these implications between the properties of the objects change when considering the set of intents concepts ordered with the Hoare (Smyth) preorder. The complexity of the lower (upper) concept lattices depends on the attribute ordered formal context and on the order relation between the attributes. To investigate the dependency of the size of the concept lattice on small/medium contexts to these parameters could be another research direction. There are many methods [1], [5] to reduce the size of the concept lattice in different settings. We want also to study how these techniques can be applied to our work.

Journal

Annals of the Alexandru Ioan Cuza University - Mathematics – de Gruyter

Published: Nov 24, 2014

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