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

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.