The possible values of critical points between varieties of lattices

TL;DR

This paper investigates the possible structures of congruence semilattices between different lattice varieties, establishing bounds on their diversity and showing the existence of many distinct classes.

Contribution

It proves the existence of bounded lattices with unique congruence structures between certain lattice varieties, and demonstrates the abundance of congruence classes in modular lattice varieties.

Findings

Existence of a bounded lattice with a unique congruence semilattice outside a given variety.

The bound of aleph 2 on the size of such lattices is optimal.

There are continuum many congruence classes of locally finite modular lattice varieties.

Abstract

We denote by Conc(L) the semilattice of all finitely generated congruences of a lattice L. For varieties (i.e., equational classes) V and W of lattices such that V is contained neither in W nor its dual, and such that every simple member of W contains a prime interval, we prove that there exists a bounded lattice A in V with at most aleph 2 elements such that Conc(A) is not isomorphic to Conc(B) for any B in W. The bound aleph 2 is optimal. As a corollary of our results, there are continuum many congruence classes of locally finite varieties of (bounded) modular lattices.