A Note on Semigroup Algebras of Permutable Semigroups

TL;DR

This paper investigates the conditions under which a specific mapping from ideals of a semigroup algebra to congruences forms a homomorphism, establishing a link with permutability of the underlying semigroup.

Contribution

It characterizes permutability of semigroups via the homomorphic property of a map related to their semigroup algebras, focusing on semilattices and rectangular bands.

Findings

The map is a homomorphism if and only if the semigroup is permutable.

The result applies specifically to semilattices and rectangular bands.

Provides a new algebraic characterization of permutability in semigroup theory.

Abstract

Let $S$ be a semigroup and $\mathbb F$ be a field. For an ideal $J$ of the semigroup algebra ${\mathbb F}[S]$ of $S$ over $\mathbb F$, let $\varrho _J$ denote the restriction (to $S$) of the congruence on ${\mathbb F}[S]$ defined by the ideal $J$. A semigroup $S$ is called a permutable semigroup if $\alpha \circ \beta =\beta \circ \alpha$ is satisfied for all congruences $\alpha$ and $\beta$ of $S$. In this paper we show that if $S$ is a semilattice or a rectangular band then $\varphi _{\{S;{\mathbb F}\}}:\ J\mapsto \varrho _J$ is a homomorphism of the semigroup $(Con ({\mathbb F}[S]);\circ )$ into the relations semigroup $({\cal B}_S; \circ )$ if and only if $S$ is a permutable semigroup.