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Growth of linear semigroups

Part of: Semigroups Chain conditions, growth conditions, and other forms of finiteness

Published online by Cambridge University Press:  09 April 2009

Jan Okniński
Jan Okniński
Affiliation:
Institute of Mathematics Warsaw UniversityBanacha 2 02-097 WarsawPoland e-mail: okninski@mimuw.edu.pl
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Abstract

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We show that the growth function of a finitely generated linear semigroup S ⊆ Mn(K) is controlled by its behaviour on finitely many cancellative subsemigroups of S. If the growth of S is polynomially bounded, then every cancellative subsemigroup T of S has a group of fractions G ⊆ Mn (K) which is nilpotent-by-finite and of finite rank. We prove that the latter condition, strengthened by the hypothesis that every such G has a finite unipotent radical, is sufficient for S to have a polynomial growth. Moreover, the degree of growth of S is then bounded by a polynomial f(n, r) in n and the maximal degree r of growth of finitely generated cancellative T ⊆ S.

MSC classification

Secondary: 20M20: Semigroups of transformations, etc. 16P90: Growth rate, Gelfand-Kirillov dimension
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 60 , Issue 1 , February 1996 , pp. 18 - 30
Copyright
Copyright © Australian Mathematical Society 1996

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