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Algebras of Ehresmann semigroups and categories. (English) Zbl 1422.20030

Semigroup Forum 95, No. 3, 509-526 (2017); erratum ibid. 96, No. 3, 603-607 (2018).
Summary: \(E\)-Ehresmann semigroups are a commonly studied generalization of inverse semigroups. They are closely related to Ehresmann categories in the same way that inverse semigroups are related to inductive groupoids. We prove that under some finiteness condition, the semigroup algebra of an \(E\)-Ehresmann semigroup is isomorphic to the category algebra of the corresponding Ehresmann category. This generalizes a result of B. Steinberg [J. Comb. Theory, Ser. A 113, No. 5, 866–881 (2006; Zbl 1148.20049)] who proved this isomorphism for inverse semigroups and inductive groupoids and a result of X. Guo and L. Chen [Proc. R. Soc. Edinb., Sect. A, Math. 142, No. 2, 371–389 (2012; Zbl 1248.20062)] who proved it for ample semigroups. We also characterize \(E\)-Ehresmann semigroups whose corresponding Ehresmann category is an EI-category and give some natural examples.

MSC:

20M10 General structure theory for semigroups
20M18 Inverse semigroups
18B40 Groupoids, semigroupoids, semigroups, groups (viewed as categories)
16S36 Ordinary and skew polynomial rings and semigroup rings

Software:

Prover9; Mace4
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References:

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