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We study simple and projective modules of a certain class of Ehresmannsemigroups, a well-studied generalization of inverse semigroups. Let $S$ be afinite right (left) restriction Ehresmann semigroup whose correspondingEhresmann category is an EI-category, that is, every endomorphism is anisomorphism. We show that the collection of finite right restriction Ehresmannsemigroups whose categories are EI is a pseudovariety. We prove that the simplemodules of the semigroup algebra $\Bbbk S$ (over any field $\Bbbk$) are formedby inducing the simple modules of the maximal subgroups of $S$ via thecorresponding Sch\"{u}tzenberger module. Moreover, we show that over fieldswith good characteristic the indecomposable projective modules can be describedin a similar way but using generalized Green's relations instead of thestandard 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 complexnumbers, we give a natural description of its indecomposable projective modulesand obtain a formula for their dimension. Moreover, we find certain zeroentries in its Cartan matrix.

Ehresmann Semigroups Whose Categories are EI and Their Representation Theory : Extended Version