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Ordered groupoid quotients and congruences on inverse semigroups

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Abstract

We introduce a preorder on an inverse semigroup S associated to any normal inverse subsemigroup N, that lies between the natural partial order and Green’s \({\mathcal {J}}\)–relation. The corresponding equivalence relation \(\simeq _N\) is not necessarily a congruence on S, but the quotient set does inherit a natural ordered groupoid structure. We show that this construction permits the factorisation of any inverse semigroup homomorphism into a composition of a quotient map and a star-injective functor, and that this decomposition implies a classification of congruences on S. We give an application to the congruence and certain normal inverse subsemigroups associate to an inverse monoid presentation.

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Authors and Affiliations

  1. Ministry of Higher Education, University of Dammam, P.O. Box 1982, Dammam, 31441, Kingdom of Saudi Arabia

    Nouf Alyamani

  2. School of Mathematical and Computer Sciences and the Maxwell Institute for the Mathematical Sciences, Heriot-Watt University, Edinburgh, EH14 4AS, UK

    N. D. Gilbert

Authors
  1. Nouf Alyamani
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  2. N. D. Gilbert
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Correspondence to N. D. Gilbert.

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Communicated by Norman R. Reilly.

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Alyamani, N., Gilbert, N.D. Ordered groupoid quotients and congruences on inverse semigroups. Semigroup Forum 96, 506–522 (2018). https://doi.org/10.1007/s00233-017-9891-4

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  • DOI: https://doi.org/10.1007/s00233-017-9891-4

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