SUB-DIRECTLY-IRREDUCIBLES OF CONGRUENCE-DISTRIBUTIVE FINITELY-GENERATED VARIETIES

Here, we prove that sub-directly-irreducibles of any congruence-distributive variety generated by a finite set of finite algebras are homomorphic images of subalgebras of members of the set, in which case they are finite, their class having a finite skeleton, and so the quasi-variety generated by the set is equational iff every meet-irreducible congruence of each member of the set is the kernel of a homomorphism from the member to a member of the set if each non-one-element subalgebra of every member of the set is simple, these results covering all finitely-generated varieties of lattice expansions. This universal elaboration is applied to the task of characterizing sub-directly-irreducibles of the intrinsic variety of Ha lkowska-Zajac' logic, solution of which results in finding the lattice of its sub-varieties, proper ones forming a four-element diamond. Likewise, as a one more application, we prove that the quasi-variety generated by any expansion of any non-Boolean four-element diamond De Morgan lattice is equational and semi-simple with sub-directly-irreducibles being exactly isomorphic copies of non-one-element subalgebras of the expansion.