The algebra of the monoid of order-preserving functions on an $n$-set and other reduced $E$-Fountain semigroups
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| Format: | Preprint |
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2024
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| _version_ | 1866929716365099008 |
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| author | Stein, Itamar |
| author_facet | Stein, Itamar |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_08075 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The algebra of the monoid of order-preserving functions on an $n$-set and other reduced $E$-Fountain semigroups Stein, Itamar Representation Theory Group Theory 20M25, 20M20, 20M30, 16G10 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. |
| title | The algebra of the monoid of order-preserving functions on an $n$-set and other reduced $E$-Fountain semigroups |
| topic | Representation Theory Group Theory 20M25, 20M20, 20M30, 16G10 |
| url | https://arxiv.org/abs/2404.08075 |