Critical points of pairs of varieties of algebras
Abstract
For a class V of algebras, denote by Conc(V) the class of all semilattices isomorphic to the semilattice Conc(A) of all compact congruences of A, for some A in V. For classes V1 and V2 of algebras, we denote by crit(V1,V2) the smallest cardinality of a semilattice in Conc(V1) which is not in Conc(V2) if it exists, infinity otherwise. We prove a general theorem, with categorical flavor, that implies that for all finitely generated congruence-distributive varieties V1 and V2, crit(V1,V2) is either finite, or alephn for some natural number n, or infinity. We also find two finitely generated modular lattice varieties V1 and V2 such that crit(V1,V2)=aleph1, thus answering a question by J. Tuma and F. Wehrung.
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