$n$元集合上保序函数幺半群及其他约化$E$-Fountain半群的代数
The algebra of the monoid of order-preserving functions on an $n$-set and other reduced $E$-Fountain semigroups
AI总结：
本文为满足广义右ample条件的约化$E$-Fountain半群构造零态射范畴，并证明半群代数与收缩范畴代数同构，统一推广多项已有结果且适用于保序函数幺半群等新例子。
摘要：
对于每个满足广义右ample条件的约化$E$-Fountain半群$S$，我们赋予其一个具有零态射的范畴$\mathcal{C}(S)$。在若干假设下，我们证明了半群代数与收缩范畴代数之间存在$\Bbbk$-代数同构$\Bbbk S\simeq\Bbbk_{0}\mathcal{C}(S)$，其中$\Bbbk$为任意含单位元的交换环。这同时推广了作者此前关于满足同余条件的约化$E$-Fountain半群的结果、Junying Guo与Xiaojiang Guo关于严格右ample半群的结果，以及Benjamin Steinberg关于幂等元居中的幂等半群的结果。我们通过两个不属于上述各类的著名幺半群展示了这一新同构的适用性：$n$元集合上的保序函数幺半群，以及带恶魔复合的二元关系幺半群。
英文摘要：
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.
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