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This article is cited in 3 scientific papers (total in 3 papers)
Finitely presented nilsemigroups: complexes with the property of uniform ellipticity
I. A. Ivanov-Pogodaevab, A. Ya. Kanel-Belovc
a Moscow Institute of Physics and Technology (National Research University), Dolgoprudny, Moscow Region
b Bar-Ilan University, Israel
c College of Mathematics and Statistics,
Shenzhen University, Shenzhen, China
Abstract:
This paper is the first in a series of three devoted to constructing a finitely presented infinite nilsemigroup satisfying the identity $x^9=0$. This solves a problem of Lev Shevrin and Mark Sapir.
In this first part we obtain a sequence of complexes formed of squares ($4$-cycles) having the following geometric properties.
1) Complexes are uniformly elliptic. A space is said to be uniformly elliptic if there is a constant $\lambda>0$ such that in the set of shortest paths of length $D$ connecting points $A$ and $B$ there are two paths such that the distance between them is at most $\lambda D$. In this case, the distance between paths with the same beginning and end is defined as the maximal distance between the corresponding points.
2) Complexes are nested.
A complex of level $n+1$ is obtained from a complex of level $n$ by adding several vertices and edges according to certain rules.
3) Paths admit local transformations. Assume that we can transform paths by replacing a path along two sides of a minimal square by the path along the other two sides. Two shortest paths with the same ends can be transformed into each other locally if these ends are vertices of a square in the embedded complex.
The geometric properties of the sequence of complexes will be further used to define finitely presented semigroups.
Keywords:
finitely presented semigroups, nilsemigroups, finitely presented rings, finitely presented groups.
Received: 08.10.2019
Revised: 01.11.2020
Citation:
I. A. Ivanov-Pogodaev, A. Ya. Kanel-Belov, “Finitely presented nilsemigroups: complexes with the property of uniform ellipticity”, Izv. Math., 85:6 (2021), 1146–1180
Linking options:
https://www.mathnet.ru/eng/im8978https://doi.org/10.1070/IM8978 https://www.mathnet.ru/eng/im/v85/i6/p126
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