Growth, unavoidable words, and M. Sapir's conjecture for semigroup varieties.
type: Item
instance of: scholarly article
title: Growth, unavoidable words, and M. Sapir's conjecture for semigroup varieties.
zbMATH Open document ID: 1039.20034
published in: Journal of Algebra
publication date: 2004-02-03T00:00:00Z
review text: Let \(u_1,u_2,\dots,u_n,\dots\) be a countable set of variables. Put \(Z_1\equiv u_1\) and \(Z_{n+1}\equiv Z_nu_{n+1}Z_n\) for any integer \(n>1\). In 1993, M.~Sapir posed the conjecture that if a nonperiodic semigroup variety \(\mathcal R\) satisfies a nontrivial identity of the type \(Z_n=W\) for some \(n>1\) then every finitely generated semigroup in \(\mathcal R\) has subexponential growth. The main result of the article under review (Theorem 1.1) states that the conjecture is true. An infinite word \(W\) is called uniformly recurrent if for every finite subword \(U\) of \(W\) there exists a number \(N\) such that every subword of \(W\) of length \(N\) contains \(U\) as a subword. Let \(S\) be a semigroup with a finite set \(X\) of generators and \(X^+\) the free semigroup over \(X\). Let \(\preceq\) denote the lexicographic order in \(X^+\). The word \(U\in X^+\) is called shortlex reduced relative to a semigroup \(S\) if for any word \(V\in X^+\) such that \(U=V\) in \(S\), we have \(\ell(U)\leq \ell(V)\) and if \(\ell(U)=\ell(V)\) then \(U\preceq V\) in \(X^+\) (where \(\ell(W)\) denotes the length of the word \(W\)). An infinite word \(\omega\) is called shortlex reduced relative to \(S\) if any finite subword of \(\omega\) is. The second result of the article (Theorem 1.2) states that if a nonperiodic semigroup variety \(\mathcal R\) satisfies a nontrivial identity of the type \(Z_n= W\) for some \(n>1\) and \(S\) is a finitely generated semigroup from \(\mathcal R\) then every infinite word which is uniformly recurrent and shortlex reduced relative to \(S\) is periodic and the length of this period is bounded above by some constant depending only on \(n\) and on the number of generators of \(S\). As applications of these results, the author constructs examples of semigroup varieties for which every noncyclic relatively free semigroup has intermediate growth and gives examples of relatively free semigroups of intermediate growth with superdimension 1.
Mathematics Subject Classification ID: 20M07
Mathematics Subject Classification ID: 20M05
zbMATH DE Number: 2037100
zbMATH Keywords: varieties of semigroups
zbMATH Keywords: growth of semigroups
zbMATH Keywords: subexponential growth, intermediate growth
zbMATH Keywords: Zimin's sequence, unavoidable words
zbMATH Keywords: relatively free semigroups
zbMATH Keywords: superdimension of semigroups
zbMATH Keywords: finitely generated semigroups
author: Shneerson, L. M.
reviewed by: Boris Vernikov
Wikidata QID: Q123300432
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2003.09.021
OpenAlex ID: W2085217910
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DOI: 10.1016/J.JALGEBRA.2003.09.021
description: scientific article; zbMATH DE number 2037100