Congruence kernels in Ockham algebras

Congruence kernels in Ockham algebras We consider, in the context of an Ockham algebra $${{\mathcal{L} = (L; f)}}$$ L = ( L ; f ) , the ideals I of L that are kernels of congruences on $${\mathcal{L}}$$ L . We describe the smallest and the largest congruences having a given kernel ideal, and show that every congruence kernel $${I \neq L}$$ I ≠ L is the intersection of the prime ideals P such that $${I \subseteq P}$$ I ⊆ P , $${P \cap f(I) = \emptyset}$$ P ∩ f ( I ) = ∅ , and $${f^{2}(I) \subseteq P}$$ f 2 ( I ) ⊆ P . The congruence kernels form a complete lattice which in general is not modular. For every non-empty subset X of L, we also describe the smallest congruence kernel to contain X, in terms of which we obtain necessary and sufficient conditions for modularity and distributivity. The case where L is of finite length is highlighted. algebra universalis Springer Journals

Congruence kernels in Ockham algebras

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Springer International Publishing
Copyright © 2017 by Springer International Publishing
Mathematics; Algebra
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