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 associated with S. We define a -module homomorphism (where is any unital commutative ring). With some assumptions, we prove that is an isomorphism of -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.