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Extensions of left regular bands by \({\mathcal {R}}\)-unipotent semigroups

Abstract

In this paper we describe \({\mathcal {R}}\)-unipotent semigroups being regular extensions of a left regular band by an \(\mathcal {R}\)-unipotent semigroup T as certain subsemigroups of a wreath product of a left regular band by T. We obtain Szendrei’s result that each E-unitary \({\mathcal {R}}\)-unipotent semigroup is embeddable into a semidirect product of a left regular band by a group. Further, specialising the first author’s notion of \(\lambda \)-semidirect product of a semigroup by a locally \({\mathcal {R}}\)-unipotent semigroup, we provide an answer to an open question raised by the authors in [Extensions and covers for semigroups whose idempotents form a left regular band, Semigroup Forum 81 (2010), 51-70].

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Acknowledgements

The authors are pleased to acknowledge the useful comments provided by the referee. These certainly contributed to an improvement of the manuscript.

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Correspondence to Paula Marques-Smith.

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Billhardt, B., Marques-Smith, P. & Martins, P.M. Extensions of left regular bands by \({\mathcal {R}}\)-unipotent semigroups. Period Math Hung (2021). https://doi.org/10.1007/s10998-021-00384-z

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Keywords

  • Embedding
  • Extension
  • Left regular band
  • R-unipotent semigroup

Mathematics Subject Classification

  • 20M10