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.
Similar content being viewed by others
REFERENCES
P. S. Novikov and S. I. Adian, Math. USSR-Izv. 2, 209–236 (1968);
Math. USSR-Izv. 2, 241–479 (1968);
Math. USSR-Izv. 2, 665–685 (1968).
S. I. Adian, Proc. Steklov Inst. Math. 289, 33–71 (2015).
Sverdlovsk Notebook: Unsolved Problems in Theory Semigroup (Ural Gos. Univ., Sverdlovsk, 1989), Vol. 3 [Russian].
O. G. Kharlampovich and M. V. Sapir, Int. J. Algebra Comput. 5 (4–5), 379–602 (1995).
I. Ivanov-Pogodaev and A. Kanel’-Belov, “Construction of infinite finitely presented nilsemigroup,” arXiv: 1412.5221.
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.”
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated by I. Ruzanova
Rights and permissions
About this article
Cite this article
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
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1064562420020027