Critical points between varieties generated by subspace lattices of vector spaces

TL;DR

This paper investigates the critical points between varieties generated by subspace lattices of vector spaces, establishing bounds and exact values for these points in various algebraic contexts.

Contribution

It introduces a method to determine the minimal cardinality of semilattices distinguishing certain algebraic varieties, especially those related to subspace lattices over finite fields.

Findings

Critical points are at least aleph_2 for certain finitely generated modular lattice varieties.

Exact critical point is aleph_2 for varieties of matrix algebras over finite fields.

Critical points between subspace lattice varieties over finite fields are precisely aleph_2.

Abstract

We denote by Conc(A) the semilattice of compact congruences of an algebra A. Given a variety V of algebras, we denote by Conc(V) the class of all semilattices isomorphic to Conc(A) for some A in V. Given varieties V1 and V2 varieties of algebras, the critical point of V1 under V2, denote by crit(V1;V2) is the smalest cardinality of a semilattice in Conc(V1) but not in Conc(V2). Given a finitely generated variety V of modular lattices, we obtain an integer l, depending of V, such that crit(V;Var(Sub F^n)) is at least aleph_2 for any n > 1 and any field F. In a second part, we prove that crit(Var(Mn);Var(Sub F^3))=aleph_2, for any finite field F and any integer n such that 1+card F< n. Similarly crit(Var(Sub F^3);Var(Sub K^3))=aleph_2, for all finite fields F and K such that card F>card K.