The Burnside ai-semiring variety defined by $x^n\approx x$

TL;DR

This paper investigates the algebraic structure of ai-semiring varieties defined by the identity x^n ≈ x, characterizing subdirectly irreducible members, and establishing properties like hereditary finite basis, countability of subvarieties, and solutions to the restricted Burnside problem.

Contribution

It provides a complete characterization of subdirectly irreducible members and properties of the ai-semiring variety ${f Sr}(n, 1)$, including hereditarily finite basis and the restricted Burnside problem.

Findings

${f Sr}(n, 1)$ is hereditarily finitely based if and only if n<4.

The lattice of subvarieties of ${f Sr}(n, 1)$ is countable if and only if n<4.

The class of locally finite members of ${f Sr}(n, 1)$ forms a variety.

Abstract

Let ${\bf Sr}(n, 1)$ denote the ai-semiring variety defined by the identity $x^n\approx x$, where $n>1$. We characterize all subdirectly irreducible members of a semisimple subvariety of ${\bf Sr}(n, 1)$. Based on this result, we prove that ${\bf Sr}(n, 1)$ is hereditarily finitely based (resp., hereditarily finitely generated) if and only if $n<4$ and that the lattice of subvarieties of ${\bf Sr}(n, 1)$ is countable if and only if $n<4$. Also, we show that the class of all locally finite members of ${\bf Sr}(n, 1)$ forms a variety and so we affirmatively answer the restricted Burnside problem for ${\bf Sr}(n, 1)$. In addition, we provide a simplified proof of the main result obtained by Gajdo\v{s} and Ku\v{r}il (Semigroup Forum 80: 92--104, 2010).