Minimal quasivarieties of semilattices over commutative groups

TL;DR

This paper classifies minimal quasivarieties of semilattices over finite abelian groups, linking them to subgroup structures, and extends the understanding to infinite groups with specific conditions.

Contribution

It establishes a one-to-one correspondence between minimal quasivarieties of semilattices over finite abelian groups and their subgroups, advancing the algebraic theory.

Findings

Minimal quasivarieties over finite abelian groups correspond to subgroups.

For infinite groups, the classification reduces to those without a zero element.

Provides a framework for understanding semilattice automorphisms over groups.

Abstract

We continue some recent investigations of W. Dziobiak, J. Jezek, and M. Maroti. Let G=(G,\cdot) be a commutative group. A semilattice over G is a semilattice enriched with G as a set of unary operations acting as semilattice automorphisms. We prove that the minimal quasivarieties of semilattices over a finite abelian group G are in one-to-one correspondence with the subgroups of G. If G is not finite, then we reduce the description of minimal quasivarieties to that of those minimal quasivarieties in which not every algebra has a zero element.