In this talk, we consider the class of finite right restriction Ehresmann semigroups whose associated Ehresmann category is an EI-category (i.e., every endomorphism is an isomorphism). We give a description of the simple modules of such semigroups using induced left Schützenberger modules. We also show that the indecomposable projective modules can be described in a similar way but using generalized Green's relations instead of the standard ones. If time allows, we will consider in greater depth a natural example - the mononid $PT_n$ of all partial functions on an $n$-element set.
This is a joint work with Stuart Margolis.