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Subsemigroups of virtually free groups: finite Malcev presentations and testing for freeness

Published online by Cambridge University Press:  03 July 2006

ALAN J. CAIN ,
EDMUND F. ROBERTSON  and
NIK RUšKUC
ALAN J. CAIN
Affiliation:
School of Mathematics and Statistics, University of St Andrews, North Haugh, St Andrews, Fife KY16 9SS. e-mail: alanc@mcs.st-andrews.ac.uk, edmund@mcs.st-andrews.ac.uk, nik@mcs.st-andrews.ac.uk
EDMUND F. ROBERTSON
Affiliation:
School of Mathematics and Statistics, University of St Andrews, North Haugh, St Andrews, Fife KY16 9SS. e-mail: alanc@mcs.st-andrews.ac.uk, edmund@mcs.st-andrews.ac.uk, nik@mcs.st-andrews.ac.uk
NIK RUšKUC
Affiliation:
School of Mathematics and Statistics, University of St Andrews, North Haugh, St Andrews, Fife KY16 9SS. e-mail: alanc@mcs.st-andrews.ac.uk, edmund@mcs.st-andrews.ac.uk, nik@mcs.st-andrews.ac.uk
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Abstract

This paper shows that, given a finite subset $X$ of a finitely generated virtually free group $F$, the freeness of the subsemigroup of $F$ generated by $X$ can be tested algorithmically. (A group is virtually free if it contains a free subgroup of finite index.) It is then shown that every finitely generated subsemigroup of $F$ has a finite Malcev presentation (a type of semigroup presentation which can be used to define any semigroup that embeds in a group), and that such a presentation can be effectively found from any finite generating set.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 141 , Issue 1 , July 2006 , pp. 57 - 66
Copyright
2006 Cambridge Philosophical Society

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