## Abstract

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 kS (over any field 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 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.

Original language | English |
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Pages (from-to) | 176-206 |

Number of pages | 31 |

Journal | Journal of Algebra |

Volume | 585 |

DOIs | |

State | Published - 1 Nov 2021 |

### Bibliographical note

Publisher Copyright:© 2021 Elsevier Inc.

## Keywords

- EI-categories
- Ehresmann semigroups
- Partial functions
- Projective modules
- Right restriction semigroups
- Semigroup algebras