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PRESENTATIONS OF INVERSE SEMIGROUPS, THEIR KERNELS AND EXTENSIONS

Part of: Semigroups

Published online by Cambridge University Press:  08 July 2011

CATARINA CARVALHO ,
ROBERT D. GRAY  and
NIK RUSKUC
CATARINA CARVALHO
Affiliation:
Centro de algebra da Universidade de Lisboa, Av. Prof. Gama Pinto 2, 1649-003 Lisboa, Portugal (email: ccarvalho@cii.fc.ul.pt)
ROBERT D. GRAY*
Affiliation:
Centro de algebra da Universidade de Lisboa, Av. Prof. Gama Pinto 2, 1649-003 Lisboa, Portugal (email: rdgray@fc.ul.pt)
NIK RUSKUC
Affiliation:
School of Mathematics and Statistics, University of St Andrews, St Andrews KY16 9SS, UK (email: nik@mcs.st-andrews.ac.uk)
*
✉For correspondence; e-mail: rdgray@fc.ul.pt
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Abstract

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Let S be an inverse semigroup and let π:S→T be a surjective homomorphism with kernel K. We show how to obtain a presentation for K from a presentation for S, and vice versa. We then investigate the relationship between the properties of S, K and T, focusing mainly on finiteness conditions. In particular we consider finite presentability, solubility of the word problem, residual finiteness, and the homological finiteness property FPn. Our results extend several classical results from combinatorial group theory concerning group extensions to inverse semigroups. Examples are also provided that highlight the differences with the special case of groups.

Keywords

inverse semigroup presentationsReidemeister–Schreierkernelfiniteness conditions

MSC classification

Secondary: 20M18: Inverse semigroups
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 90 , Issue 3 , June 2011 , pp. 289 - 316
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2011

Footnotes

Carvalho was supported by FCT grant SFRH/BPD/26216/2006. Part of this work was done while Gray was an EPSRC Postdoctoral Research Fellow at the University of St Andrews, Scotland. Gray was also partially supported by FCT and FEDER, project POCTI-ISFL-1-143 of the Centro de Álgebra da Universidade de Lisboa, and by the project PTDC/MAT/69514/2006.

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