Critical Points of Pairs of Varieties of Algebras

For a class V of algebras, denote by Conc V the class of all (∨, 0)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 (∨, 0)-semilattice in Conc V1 which is not in Conc V2 if it exists, ∞ otherwise. We prove a general theorem, with categorical flavor, that implies that for all finitely generated congruencedistributive varieties V1 and V2, crit(V1;V2) is either finite, or אn for some natural number n, or ∞. We also find two finitely generated modular lattice varieties V1 and V2 such that crit(V1;V2) = א1, thus answering a question by J. Tůma and F. Wehrung.