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GROUPS AND SEMIGROUPS WITH A ONE-COUNTER WORD PROBLEM

Part of: Special aspects of infinite or finite groups Computability and recursion theory

Published online by Cambridge University Press:  01 October 2008

DEREK F. HOLT ,
MATTHEW D. OWENS  and
RICHARD M. THOMAS
DEREK F. HOLT*
Affiliation:
Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK (email: D.F.Holt@warwick.ac.uk)
MATTHEW D. OWENS
Affiliation:
Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK
RICHARD M. THOMAS
Affiliation:
Department of Computer Science, University of Leicester, Leicester LE1 7RH, UK (email: rmt@mcs.le.ac.uk)
*
✉For correspondence; e-mail: D.F.Holt@warwick.ac.uk
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Abstract

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We prove that a finitely generated semigroup whose word problem is a one-counter language has a linear growth function. This provides us with a very strong restriction on the structure of such a semigroup, which, in particular, yields an elementary proof of a result of Herbst, that a group with a one-counter word problem is virtually cyclic. We prove also that the word problem of a group is an intersection of finitely many one-counter languages if and only if the group is virtually abelian.

Keywords

groupssemigroupsword problem

MSC classification

Secondary: 20F10: Word problems, other decision problems, connections with logic and automata 03D40: Word problems, etc.
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 85 , Issue 2 , October 2008 , pp. 197 - 209
Copyright
Copyright © 2008 Australian Mathematical Society

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