The possible values of critical points between strongly congruence-proper varieties of algebras

TL;DR

This paper investigates the possible values of critical points between strongly congruence-proper varieties of algebras, establishing that if certain conditions are met, the difference in their congruence semilattices can be as large as aleph 2, with the bound being optimal.

Contribution

It extends the understanding of critical points between algebraic varieties by providing bounds and conditions under which large differences in congruence semilattices occur.

Findings

Existence of a semilattice of size aleph 2 in the difference of congruence semilattices.

Extension of results to quasivarieties with finitely many relation symbols.

Sharpness of the aleph 2 bound.

Abstract

We denote by Conc(A) the semilattice of all finitely generated congruences of an (universal) algebra A, and we define Conc(V) as the class of all isomorphic copies of all Conc(A), for A in V, for any variety V of algebras. Let V and W be locally finite varieties of algebras such that for each finite algebra A in V there are, up to isomorphism, only finitely many B in W such that A and B have isomorphic congruence lattices, and every such B is finite. If Conc(V) is not contained in Conc(W), then there exists a semilattice of cardinality aleph 2 in Conc(V)-Conc(W). Our result extends to quasivarieties of first-order structures, with finitely many relation symbols, and relative congruence lattices. In particular, if W is a finitely generated variety of algebras, then this occurs in case W omits the tame congruence theory types 1 and 5; which, in turn, occurs in case W satisfies a nontrivial congruence identity. The bound aleph 2 is sharp.