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Algebras of Ehresmann semigroups and categories

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An Erratum to this article was published on 21 August 2017

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Abstract

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 Steinberg who proved this isomorphism for inverse semigroups and inductive groupoids and a result of Guo and Chen 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.

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  • 21 August 2017

    An erratum to this article has been published.

Notes

  1. This part of the proof is actually due to Michael Kinyon who proved it using the automated theorem prover called Prover9. For general information on Prover9 see [17].

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Acknowledgements

The author is grateful to Michael Kinyon for proving that \({{\mathrm{Reg}}}_E(S)\) is a subsemigroup (the main part of Lemma 5.15). The author also thanks the referee for his\her helpful comments.

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Authors and Affiliations

  1. Department of Mathematics, Bar Ilan University, 52900, Ramat Gan, Israel

    Itamar Stein

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  1. Itamar Stein
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Correspondence to Itamar Stein.

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Communicated by Victoria Gould.

This paper is part of the author’s PHD thesis, being carried out under the supervision of Prof. Stuart Margolis. The author’s research was supported by Grant No. 2012080 from the United States-Israel Binational Science Foundation (BSF) and by the Israeli Ministry of Science, Technology and Space.

An erratum to this article is available at https://doi.org/10.1007/s00233-017-9895-0.

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Stein, I. Algebras of Ehresmann semigroups and categories. Semigroup Forum 95, 509–526 (2017). https://doi.org/10.1007/s00233-016-9838-1

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