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Higher commutators in semigroups with zero

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Abstract

We introduce the notion of the higher commutator of ideals in semigroups. For semigroups with zero, it is shown that the higher order commutator of Rees congruences is equal to the Rees congruence of the commutator of the corresponding ideals. We obtain that, for Rees congruences, higher order commutator is a composition of binary commutators. As a consequence, we prove that in semigroups with zero all four conditions of solvability, supernilpotency, nilpotency and nilpotency in the sense of semigroup theory, are equivalent.

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Acknowledgements

The authors thank the referees for drawing our attention that the Lemma 2.8 is true in general and for other valuable suggestions.

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Authors and Affiliations

  1. Department of Mathematics, University of East Sarajevo, 71123, East Sarajevo, Bosnia and Herzegovina

    Jelena Radović

  2. Department of Mathematics and Informatics, Faculty of Sciences, University of Novi Sad, 21000, Novi Sad, Serbia

    Jelena Radović & Nebojša Mudrinski

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  1. Jelena Radović
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  2. Nebojša Mudrinski
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Correspondence to Jelena Radović.

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Presented by P. Mayr.

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N. Mudrinski has been supported by the Ministry of Science, Education and Technological Development of the Republic of Serbia (Grant No. 451-03-68/2022-14/200125).

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Radović, J., Mudrinski, N. Higher commutators in semigroups with zero. Algebra Univers. 84, 12 (2023). https://doi.org/10.1007/s00012-023-00809-5

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  • DOI: https://doi.org/10.1007/s00012-023-00809-5

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