Ehresmann semigroups whose categories are EI and their representation theory. (English) Zbl 1509.20116
Summary: 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.
MSC:
| 20M18 | Inverse semigroups |
| 20M30 | Representation of semigroups; actions of semigroups on sets |
| 16G10 | Representations of associative Artinian rings |
Keywords:
Ehresmann semigroups; EI-categories; right restriction semigroups; semigroup algebras; projective modules; partial functionsReferences:
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