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On congruence lattices of nilsemigroups

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Abstract

We prove that the congruence lattice of a nilsemigroup is modular if and only if the width of the semigroup, as a poset, is at most two, and distributive if and only if its width is one. In the latter case, such semigroups therefore coincide with the nil \(\Delta \)-semigroups. It is further shown that if a finitely generated nilsemigroup has modular congruence lattice, then the semigroup is finite.

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Acknowledgements

The authors are grateful to Vladimir Repnitskii, for his attention to the paper, and to the referee for suggestions that improved its exposition. Alexander L. Popovich acknowledges support from the Presidential Programme “Leading Scientific Schools of the Russian Federation”, Project No. 5161.2014.1, the Russian Foundation for Basic Research, Project No. 14-01-00524, the Ministry of Education and Science of the Russian Federation, Project No. 1.1999.2014/K, and the Competitiveness Program of Ural Federal University.

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, Ural Federal University, Ekaterinburg, Russia, 620000

    Alexander L. Popovich

  2. Department of Mathematics, Statistics and Computer Science, Marquette University, Milwaukee, WI, 53201, USA

    Peter R. Jones

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  1. Alexander L. Popovich
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  2. Peter R. Jones
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Correspondence to Peter R. Jones.

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Communicated by Mikhail Volkov.

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Popovich, A.L., Jones, P.R. On congruence lattices of nilsemigroups. Semigroup Forum 95, 314–320 (2017). https://doi.org/10.1007/s00233-016-9837-2

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  • DOI: https://doi.org/10.1007/s00233-016-9837-2

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