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

Abstract With every reduced E -Fountain semigroup S which satisfies the generalized right ample condition we associate a category with zero morphisms $$\mathcal {C}(S)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>C</mml:mi> <mml:mo>(</mml:mo> <mml:mi>S</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . Under some assumptions we prove an isomorphism of $$\Bbbk $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>k</mml:mi> </mml:math> -algebras $$\Bbbk S\simeq \Bbbk _{0}\mathcal {C}(S)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>k</mml:mi> <mml:mi>S</mml:mi> <mml:mo>≃</mml:mo> <mml:msub> <mml:mi>k</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mi>C</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>S</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> between the semigroup algebra and the contracted category algebra where $$\Bbbk $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>k</mml:mi> </mml:math> 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.