论文标题
Ehresmann的类别是EI及其表示理论：扩展版本
Ehresmann Semigroups Whose Categories are EI and Their Representation Theory : Extended Version
论文作者
论文摘要
我们研究了某些类别的Ehresmann Semigroups的简单和投射模块，这是对反向半群的概括的概括。令$ s $为有限的右（左）限制Ehresmann Semigroup，其相应的Ehresmann类别是EI类别，也就是说，每个内态性都是同构。我们表明，有限的权利限制的收集Ehresmann Semigroups的类别是EI是伪动力的。我们证明，通过相应的schützenberger模块诱导$ s $的最大亚组的简单模块，形成了Semigroup代数$ \ bbbk s $（在任何字段$ \ bbbk $上）的简单模块。此外，我们表明，在具有良好特征的字段上，可以以类似的方式来描述不可分解的投影模块，但使用广义格林的关系而不是标准关系。作为一个自然的示例，我们考虑了所有部分功能在$ n $ element set上的所有部分功能的monoid $ \ natercal {pt} _ {n} $。在复数的领域，我们对其不可分解的投影模块进行了自然描述，并获得了其尺寸的公式。此外，我们在其Cartan矩阵中发现了某些零条目。
We study simple and projective modules of a certain class of Ehresmann semigroups, a well-studied generalization of inverse semigroups. Let $S$ be a finite right (left) restriction Ehresmann semigroup whose corresponding Ehresmann category is an EI-category, that is, every endomorphism is an isomorphism. We show that the collection of finite right restriction Ehresmann semigroups whose categories are EI is a pseudovariety. We prove that the simple modules of the semigroup algebra $\Bbbk S$ (over any field $\Bbbk$) are formed by inducing the simple modules of the maximal subgroups of $S$ via the corresponding Schützenberger module. Moreover, we show that over fields with good characteristic the indecomposable projective modules can be described in a similar way but using generalized Green's relations instead of the standard ones. As a natural example, we consider the monoid $\mathcal{PT}_{n}$ of all partial functions on an $n$-element set. Over the field of complex numbers, we give a natural description of its indecomposable projective modules and obtain a formula for their dimension. Moreover, we find certain zero entries in its Cartan matrix.
