## 2007.176: On the rational subset problem for groups

2007.176:
Mark Kambites, Pedro V. Silva and Benjamin Steinberg
(2006)
*On the rational subset problem for groups.*
Journal of Algebra, 309 (2).
pp. 622-639.
ISSN 0021-8693

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DOI: 10.1016/j.jalgebra.2006.05.020

## Abstract

We use language theory to study the rational subset problem for groups and monoids. We show that the decidability of this problem is preserved under graph of groups constructions with finite edge groups. In particular, it passes through free products amalgamated over finite subgroups and HNN extensions with finite associated subgroups. We provide a simple proof of a result of Grunschlag showing that the decidability of this problem is a virtual property. We prove further that the problem is decidable for a direct product of a group G with a monoid M if and only if membership is uniformly decidable for G-automaton subsets of M. It follows that a direct product of a free group with any abelian group or commutative monoid has decidable rational subset membership.

Item Type: | Article |
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Uncontrolled Keywords: | Group; Formal language; Decision problem; Rational subset |

Subjects: | MSC 2000 > 20 Group theory and generalizations |

MIMS number: | 2007.176 |

Deposited By: | Mrs Louise Healey |

Deposited On: | 19 November 2007 |

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