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Every quasitrivial n-ary semigroup is reducible to a semigroup

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Abstract

We show that every quasitrivial n-ary semigroup is reducible to a binary semigroup, and we provide necessary and sufficient conditions for such a reduction to be unique. These results are then refined in the case of symmetric n-ary semigroups. We also explicitly determine the sizes of these classes when the semigroups are defined on finite sets. As a byproduct of these enumerations, we obtain several new integer sequences.

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Acknowledgements

Both authors would like to thank Jean-Luc Marichal and the anonymous referee for their useful comments and insightful remarks that helped improving the current paper.

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Authors and Affiliations

  1. Université de Lorraine, CNRS, Inria, LORIA, 54000, Nancy, France

    Miguel Couceiro

  2. Mathematics Research Unit, University of Luxembourg, Maison du Nombre, 6, avenue de la Fonte, 4364, Esch-sur-Alzette, Luxembourg

    Jimmy Devillet

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  1. Miguel Couceiro
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  2. Jimmy Devillet
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Correspondence to Miguel Couceiro.

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Jimmy Devillet is supported by the Luxembourg National Research Fund under the project PRIDE 15/10949314/GSM.

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Couceiro, M., Devillet, J. Every quasitrivial n-ary semigroup is reducible to a semigroup. Algebra Univers. 80, 51 (2019). https://doi.org/10.1007/s00012-019-0626-0

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  • DOI: https://doi.org/10.1007/s00012-019-0626-0

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