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 $\mathbb{k}S$ (over any field $\mathbb{k}$) 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.

我们研究了某一类 Ehresmann 半群的简单和射影模，这是对逆半群的深入研究。设S是一个有限右（左）限制 Ehresmann 半群，其对应的 Ehresmann 范畴是一个 EI 范畴，即每个自同构都是一个同构。我们证明了类别为 EI 的有限右限制 Ehresmann 半群的集合是一个伪变体。我们证明了半群代数的简单模$\mathbb{克}秒$ （在任何领域 $\mathbb{克}$) 是通过相应的 Schützenberger 模归纳S的最大子群的简单模而形成的。此外，我们表明，在具有良好特性的领域上，不可分解的射影模块可以用类似的方式描述，但使用广义格林关系而不是标准关系。作为一个自然的例子，我们考虑幺半群${\mathcal{PT}}_n$n元素集上的所有偏函数。在复数领域，我们对其不可分解的射影模进行了自然的描述，并获得了它们的维数公式。此外，我们在其 Cartan 矩阵中找到了某些零项。