Critical points of pairs of varieties of algebras

TL;DR

This paper investigates the minimal sizes of semilattices that distinguish certain algebraic varieties, establishing a classification of these sizes and providing examples that answer longstanding questions.

Contribution

It proves a general theorem classifying critical points between finitely generated congruence-distributive varieties and constructs examples with critical point aleph_1.

Findings

Critical points are either finite, aleph_n, or infinite for certain varieties.

Established a categorical framework for analyzing critical points.

Provided examples of varieties with critical point aleph_1.

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 aleph_n for some natural number n, or infinity. We also find two finitely generated modular lattice varieties V1 and V2 such that crit(V1,V2)=aleph_1, thus answering a question by J. Tuma and F. Wehrung.