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 partial composition $\mathcal{C}(S)$. Under some assumptions we prove an isomorphism of $\Bbbk$-algebras $\Bbbk S\simeq\Bbbk\mathcal{C}(S)$ between the semigroup algebra and the 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 $\mathcal{O}_{n}$ of order-preserving functions on an $n$-set and the monoid of binary relations with demonic composition.
- Publication:
-
arXiv e-prints
- Pub Date:
- April 2024
- DOI:
- 10.48550/arXiv.2404.08075
- arXiv:
- arXiv:2404.08075
- Bibcode:
- 2024arXiv240408075S
- Keywords:
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- Mathematics - Representation Theory;
- Mathematics - Group Theory;
- 20M25;
- 20M20;
- 20M30;
- 16G10
- E-Print:
- 24 pages