Embedding lattices of quasivarieties of periodic groups into lattices of additively idempotent semiring varieties: An algebraic proof

TL;DR

This paper provides a new algebraic proof of the embedding of quasivarieties of periodic groups into lattices of additively idempotent semiring varieties, revealing significant differences in lattice properties for various exponents.

Contribution

It offers a direct algebraic proof of an existing embedding result and introduces a new identity basis for the top variety within the lattice.

Findings

The lattice properties for n≥3 differ greatly from those for n=1 or 2.

A new identity basis for the top variety of the interval is established.

The properties of the lattice L(Sr_n) vary significantly with n.

Abstract

A general result by Jackson (Flat algebras and the translation of universal Horn logic to equational logic, J. Symb. Log. 73(1) (2008) 90--128) implies that the lattice of all quasivarieties of groups of exponent dividing $n$ embeds into the lattice $L(\mathbf{Sr}_n)$ of all varieties of additively idempotent semirings whose multiplicative semigroups are unions of groups of exponent dividing $n$; the image of this embedding is an interval in $L(\mathbf{Sr}_n)$. We provide a new, direct, and purely algebraic proof of these facts and present a new identity basis for the top variety of the interval. In addition, we obtain new information about the lattice $L(\mathbf{Sr}_n)$, demonstrating that the properties of the lattice for $n\ge 3$ differ drastically from those previously known when $n=1$ or $2$.