CRITICAL POINTS OF PAIRS OF VARIETIES OF ALGEBRAS
Abstract
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 ℵn 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.
This paper is a part of the author's "Doctorat de l'université de Caen", prepared under the supervision of Friedrich Wehrung.
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