Rational growth of a class of inverse semigroups
type: Item
instance of: scholarly article
title: Rational growth of a class of inverse semigroups
zbMATH Open document ID: 0910.20043
author: Joseph Lau
published in: Journal of Algebra
publication date: 1999-04-19T00:00:00Z
review text: This is the first of two consecutive papers in this journal (for the second paper see the following review Zbl 0910.20044) on growth functions of inverse semigroups. Let \(S\) be a semigroup generated by a finite set \(X\) and let \(\ell(t)\) denote the minimal length of \(t\in S\) over \(X\). The author works with the power series \(h_S(z)\) where the coefficient of \(z^m\) is \(h_S(m)=|\{t\in S:\ell(t)=m\}|\). The semigroup \(S\) is said to have rational growth if \(h_S(z)\) is a rational function of \(z\). We say that \(S\) has polynomial growth [resp. exponential growth] if \(h_S(m)\) is bounded above by \(m^d\) (some \(d\geq 1)\) [resp. bounded below by \(\alpha^m\) (some \(\alpha>1\))]. The author first proves in a new way a known result: the free inverse semigroup on a finite generating set of order at least two has irrational growth. A condition for a Rees quotient \(S\) of a finitely generated free inverse semigroup in terms of restrictions on cycles of a certain graph associated with \(S\) is sufficient to ensure rational growth. It is proved that, within this class of semigroups, polynomial growth implies rational growth, but the converse is false.
Mathematics Subject Classification ID: 20M18
Mathematics Subject Classification ID: 20M05
zbMATH DE Number: 1218667
zbMATH Keywords: growth functions
zbMATH Keywords: inverse semigroups
zbMATH Keywords: power series
zbMATH Keywords: rational growth
zbMATH Keywords: polynomial growth
zbMATH Keywords: exponential growth
zbMATH Keywords: finite generating sets
zbMATH Keywords: finitely generated free inverse semigroups
reviewed by: Peter M. Higgins
MaRDI profile type: Publication
full work available at URL: jabr.1997.7267
OpenAlex ID: W2091354237
cites work: GROWTH AND EXISTENCE OF IDENTITIES IN A CLASS OF FINITELY PRESENTED INVERSE SEMIGROUPS WITH ZERO
cites work: GROWTH OF FINITELY PRESENTED REES QUOTIENTS OF FREE INVERSE SEMIGROUPS
cites work: Q4003180
DOI: 10.1006/JABR.1997.7267
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description: scientific article; zbMATH DE number 1218667