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Construction of Infinite Finitely Presented Nilsemigroup

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Abstract

An infinite finitely presented nilsemigroup with identity x9 = 0 is constructed. This construction answers the question of L.N. Shevrin and M.V. Sapir. The proof is based on the construction of a sequence of geometric complexes, each obtained by gluing several simple 4-cycles (squares). These complexes have certain geometric and combinatorial properties. Actually, the semigroup is the set of word codings of paths on such complexes. Each word codes a path on some complex. Defining relations correspond to pairs of equivalent short paths. The shortest paths in terms of the natural metric are associated with nonzero words in the subgroup. Codings that are not presented by some path or presented by non-shortest paths can be reduced to a zero word.

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ACKNOWLEDGMENTS

We are grateful to V.N. Latyshev and A.V. Mikhalev, heads of the seminar “Ring theory” in the Higher Algebra Department of the Faculty of Mechanics and Mathematics of Moscow State University, for helpful discussions and long-lasting interest in our work. We are also grateful to I.A. Rips, L.N. Shevrin, A.H. Shen, N.K. Vereshchagin, A. Ershler for helpful discussions, to P. Durand, Y. Sella, L.A. Bokut, Y. Chen, T. Fernique for their encouragement of conference participation, and to A.S. Malistov for his assistance in preparing the manuscript. Special gratitude is expressed to A.L. Semenov for his helpful advice and interest in this work.

Funding

This work was supported by the Russian Science Foundation, grant no. 17-11-01377. The second author is the winner of the contest “Young Russian Mathematics.”

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Authors and Affiliations

  1. Shenzhen University, Shenzhen, China

    A. Ya. Belov-Kanel

  2. Bar-Ilan University, Ramat-Gan, Israel

    A. Ya. Belov-Kanel

  3. Moscow Institute of Physics and Technology (National Research University), Dolgoprudnyi, Moscow oblast, Russia

    I. A. Ivanov-Pogodaev

Authors
  1. A. Ya. Belov-Kanel
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  2. I. A. Ivanov-Pogodaev
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Correspondence to I. A. Ivanov-Pogodaev.

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Translated by I. Ruzanova

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Belov-Kanel, A.Y., Ivanov-Pogodaev, I.A. Construction of Infinite Finitely Presented Nilsemigroup. Dokl. Math. 101, 81–85 (2020). https://doi.org/10.1134/S1064562420020027

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  • DOI: https://doi.org/10.1134/S1064562420020027

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