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

Itamar Stein

TL;DR

The paper develops a unified isomorphism between semigroup algebras and contracted category algebras for reduced $E$-Fountain semigroups under a generalized right ample condition, extending and unifying prior results. By introducing the contracted category $\mathcal{C}(S)$ with zero morphisms and a generalized composition, it shows $\Bbbk S \cong \Bbbk_{0}\mathcal{C}(S)$ under principled finiteness, and it recovers classical cases when the right congruence or congruence conditions hold. The framework encompasses new examples, notably the monoid of order-preserving functions $\mathcal{O}_{n}$ and the demonic composition on binary relations, with explicit descriptions of morphisms, kernels, and semisimple images. The results yield generalized matrix representations and central-idempotent decompositions, connecting to established theories (Morita equivalence, Dold–Kan) and offering concrete tools for representation-theoretic analysis of semigroup algebras. Overall, the work broadens the scope of algebraic and categorical methods in semigroup theory, providing a versatile bridge between semigroups and category algebras across several classical and new examples.

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.

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

TL;DR

The paper develops a unified isomorphism between semigroup algebras and contracted category algebras for reduced -Fountain semigroups under a generalized right ample condition, extending and unifying prior results. By introducing the contracted category with zero morphisms and a generalized composition, it shows under principled finiteness, and it recovers classical cases when the right congruence or congruence conditions hold. The framework encompasses new examples, notably the monoid of order-preserving functions and the demonic composition on binary relations, with explicit descriptions of morphisms, kernels, and semisimple images. The results yield generalized matrix representations and central-idempotent decompositions, connecting to established theories (Morita equivalence, Dold–Kan) and offering concrete tools for representation-theoretic analysis of semigroup algebras. Overall, the work broadens the scope of algebraic and categorical methods in semigroup theory, providing a versatile bridge between semigroups and category algebras across several classical and new examples.

Abstract

With every reduced -Fountain semigroup which satisfies the generalized right ample condition we associate a category with zero morphisms . Under some assumptions we prove an isomorphism of -algebras between the semigroup algebra and the contracted 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 of order-preserving functions on an -set and the monoid of binary relations with demonic composition.
Paper Structure (12 sections, 27 theorems, 68 equations)

This paper contains 12 sections, 27 theorems, 68 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Semigroups and categories
  4. Algebras
  5. An isomorphism theorem and the generalized right ample condition
  6. Generalization of older results
  7. The congruence condition
  8. Generalized matrix representations and strict right ample semigroups
  9. Finite idempotent semigroups with central idempotents
  10. Additional examples
  11. The monoid of order-preserving functions
  12. Binary relations with demonic composition

Key Result

Lemma 2.3

Let $S$ be a reduced $E$-Fountain semigroup and let $a,b\in S$. Then $(ba)^{\ast}\leq a^{\ast}$ and $(ba)^{+}\leq b^{+}$.

Theorems & Definitions (57)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Definition 2.4
  • Definition 3.1
  • Remark 3.2
  • Proposition 3.3
  • proof
  • Definition 3.4
  • ...and 47 more