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

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 $\mathbb{k}$-module homomorphism $\phi :\mathbb{k}S\to \mathbb{k}\mathcal{C}$ (where $\mathbb{k}$ is any unital commutative ring). With some assumptions, we prove that $\phi$ is an isomorphism of $\mathbb{k}$-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.

21 pages