Narrow Results

For a class of algebras, denote by Con_c the class of all (∨, 0)-semilattices isomorphic to the semilattice Con_cA of all compact congruences of A, for some A in . For classes and of algebras, we denote by the smallest cardinality of a (∨, 0)-semilattices in Con_c which is not in Con_c if it exists, ∞ otherwise. We prove a general theorem, with categorical flavor, that implies that for all finitely generated congruence-distributive varieties and , is either finite, or ℵ_n for some natural number n, or ∞. We also find two finitely generated modular lattice varieties and such that , thus answering a question by J. Tůma and F. Wehrung.