Critical points of pairs of varieties of algebras

International Journal of Algebra and Computation, 19(1) (2009), 1-40
http://dx.doi.org/10.1142/S0218196709004932

For a class of algebras, denote by Conc the class of all (?, 0)-semilattices isomorphic to the semilattice ConcA 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 Conc which is not in Conc 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 ?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. T?ma and F. Wehrung.