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Free Monoids are Coherent

Part of: Semigroups

Published online by Cambridge University Press:  15 June 2016

Victoria Gould ,
Miklós Hartmann  and
Nik Ruškuc
Victoria Gould
Affiliation:
Department of Mathematics, University of York, Heslington, York YO10 5DD, UK (victoria.gould@york.ac.uk)
Miklós Hartmann
Affiliation:
Department of Mathematics, University of Szeged, Aradi vertanuk tere 1, H-6720 Szeged, Hungary (hartm@math.u-szeged.hu)
Nik Ruškuc
Affiliation:
School of Mathematics and Statistics, University of St Andrews, St Andrews, Fife KY16 9SS, UK (nik@mcs.st-and.ac.uk)
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Abstract

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A monoid S is said to be right coherent if every finitely generated subact of every finitely presented right S-act is finitely presented. Left coherency is defined dually and S is coherent if it is both right and left coherent. These notions are analogous to those for a ring R (where, of course, S-acts are replaced by R-modules). Choo et al. have shown that free rings are coherent. In this paper we prove that, correspondingly, any free monoid is coherent, thus answering a question posed by Gould in 1992.

Keywords

free monoidsS-actscoherency

MSC classification

Secondary: 20M05: Free semigroups, generators and relations, word problems 20M30: Representation of semigroups; actions of semigroups on sets
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 60 , Issue 1 , February 2017 , pp. 127 - 131
Copyright
Copyright © Edinburgh Mathematical Society 2016