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Inductive groupoids and cross-connections of regular semigroups

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Abstract

There are two major structure theorems for an arbitrary regular semigroup using categories, both due to Nambooripad. The first construction using inductive groupoids departs from the biordered set structure of a given regular semigroup. This approach belongs to the realm of the celebrated Ehresmann–Schein–Nambooripad Theorem and its subsequent generalisations. The second construction is a generalisation of Grillet’s work on cross-connected partially ordered sets, arising from the principal ideals of the given semigroup. In this article, we establish a direct equivalence between these two seemingly different constructions. We show how the cross-connection representation of a regular semigroup may be constructed directly from the inductive groupoid of the semigroup, and vice versa.

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Acknowledgements

We are very grateful to J. Meakin, University of Nebraska-Lincoln, for reading an initial draft of the article and helping us with several enlightening suggestions. We also thank the referee for the careful reading of the article and the detailed comments which helped us to improve the manuscript.

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  1. Institute of Natural Sciences and Mathematics, Ural Federal University, 620000, Ekaterinburg, Russia

    P. A. Azeef Muhammed & M. V. Volkov

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  1. P. A. Azeef Muhammed
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Correspondence to P. A. Azeef Muhammed.

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Dedicated to Mária B. Szendrei, one of the leading contributors to the field of regular semigroups, on the occasion of her birthday.

The authors acknowledge the financial support by the Competitiveness Enhancement Program of the Ural Federal University. They were also supported by the Russian Foundation for Basic Research, grant no. 17-01-00551, and the Ministry of Education and Science of the Russian Federation, project no. 1.3253.2017.

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Muhammed, P.A.A., Volkov, M.V. Inductive groupoids and cross-connections of regular semigroups. Acta Math. Hungar. 157, 80–120 (2019). https://doi.org/10.1007/s10474-018-0888-6

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