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 partial composition C(S). Under some assumptions we prove an isomorphism of -algebras S(S) between the semigroup algebra and the category algebra where 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 Oₙ of order-preserving functions on an n-set and the monoid of binary relations with demonic composition.