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

With every reduced E -Fountain semigroup S which satisfies the generalized right ample condition we associate a category with zero morphisms C(S) C ( S ) . Under some assumptions we prove an isomorphism of k -algebras S ₀C(S) k S ≃ k 0 C ( S ) between the semigroup algebra and the contracted category algebra where k 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.