Algebras of Reduced $E$-Fountain Semigroups and the Generalized Ample Identity

TL;DR

This paper explores the algebraic structure of reduced $E$-Fountain semigroups, establishing conditions under which their semigroup algebras are isomorphic to certain category algebras, generalizing previous results.

Contribution

It introduces a new construction linking reduced $E$-Fountain semigroups satisfying the congruence condition to associated categories, and characterizes algebra isomorphisms via a generalized ample identity.

Findings

The homomorphism $$ is an isomorphism under a weak right ample identity.

Provides a unified framework generalizing previous semigroup algebra results.

Connects semigroup properties with categorical algebra structures.

Abstract

Let $S$ be a reduced $E$-Fountain semigroup. If $S$ satisfies the congruence condition, there is a natural construction of a category $\mathcal{C}$ associated with $S$. We define a $\Bbbk$-module homomorphism $\varphi:\Bbbk S\to\Bbbk\mathcal{C}$ (where $\Bbbk$ is any unital commutative ring). With some assumptions, we prove that $\varphi$ is an isomorphism of $\Bbbk$-algebras if and only if some weak form of the right ample identity holds in $S$. This gives a unified generalization for a result of the author on right restriction $E$-Ehresmann semigroups and a result of Margolis and Steinberg on the Catalan monoid.