Journal of the Australian Mathematical Society

Table of Contents - June 2011 - Volume 90, Issue 03  

Research Article

PRESENTATIONS OF INVERSE SEMIGROUPS, THEIR KERNELS AND EXTENSIONS

CATARINA CARVALHOa1, ROBERT D. GRAYa2 c1 and NIK RUSKUCa3

a1 Centro de algebra da Universidade de Lisboa, Av. Prof. Gama Pinto 2, 1649-003 Lisboa, Portugal (email: ccarvalho@cii.fc.ul.pt)

a2 Centro de algebra da Universidade de Lisboa, Av. Prof. Gama Pinto 2, 1649-003 Lisboa, Portugal (email: rdgray@fc.ul.pt)

a3 School of Mathematics and Statistics, University of St Andrews, St Andrews KY16 9SS, UK (email: nik@mcs.st-andrews.ac.uk)

Abstract

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.

(Received May 20 2010)

(Accepted October 28 2010)

(Online publication July 08 2011)

2010 Mathematics subject classification

  • primary 20M18; secondary 20M05

Keywords and phrases

  • inverse semigroup presentations;
  • Reidemeister–Schreier;
  • kernel;
  • finiteness conditions

Correspondence:

c1 For correspondence; e-mail: rdgray@fc.ul.pt

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.

Communicated by M. G. Jackson