The algebra of the monoid of order-preserving functions on an $n$-set and other reduced $E$-Fountain semigroups

TL;DR

This paper establishes an algebraic isomorphism linking semigroup algebras of certain reduced E-Fountain semigroups to contracted category algebras, generalizing previous results and applying to well-known monoids like order-preserving functions.

Contribution

It introduces a new isomorphism between semigroup and category algebras for reduced E-Fountain semigroups satisfying the generalized right ample condition, extending prior work.

Findings

Proves algebra isomorphism for specific semigroups and categories.

Demonstrates applicability to monoids of order-preserving functions and binary relations.

Generalizes previous algebraic results in semigroup theory.

Abstract

With every reduced $E$-Fountain semigroup $S$ which satisfies the generalized right ample condition we associate a category with zero morphisms $\mathcal{C}(S)$. Under some assumptions we prove an isomorphism of $\Bbbk$-algebras $\Bbbk S\simeq\Bbbk_{0}\mathcal{C}(S)$ between the semigroup algebra and the contracted category algebra where $\Bbbk$ is any commutative unital ring. This is a simultaneous generalization of a former result of the author on reduced E-Fountain semigroups which satisfy the congruence condition, a result of Junying Guo and Xiaojiang Guo on strict right ample semigroups and a result of Benjamin Steinberg on idempotent semigroups with central idempotents. The applicability of the new isomorphism is demonstrated with two well-known monoids which are not members of the above classes. The monoid of order-preserving functions on an $n$-set and the monoid of binary relations with demonic composition.