Congruence coherent double MS-algebras

Abstract

If A is an algebra and ϑ is a congruence on A then A is said to be ϑ-coherent provided that, for every subalgebra B of A, if B contains some ϑ-class then B is a union of ϑ-classes. An algebra A is said to be congruence coherent if it is ϑ-coherent for every ϑ∈>ConA. This notion was investigated by Beazer [2] in the context of de Morgan algebras. Specifically, he showed that a de Morgan algebra is congruence coherent if and only if it is boolean, or simple, or the 4-element de Morgan chain. He also showed that if an algebra in the Berman class K_1,1 of Ockham algebras is congruence coherent then it is necessarily a de Morgan algebra; and that a p-algebra is congruence coherent if and only if it is boolean. This notion has also been considered in the context of distributive double p-algebras by Adams, Atallah and Beazer [1] who showed that particular examples of congruence coherent double p-algebras are those that are congruence regular (in the sense that if two congruences have a class in common then they coincide). In this paperNATO Collaborative Research Grant 960153 is gratefully acknowledged. we extend the results of Beazer to the class of double MS-algebras.