Hostname: page-component-76fb5796d-vfjqv Total loading time: 0 Render date: 2024-04-29T19:49:38.844Z Has data issue: false hasContentIssue false
  • English
  • Français

HNN extensions of inverse semigroups with zero

Published online by Cambridge University Press:  12 February 2007

E. R. DOMBI  and
N. D. GILBERT
E. R. DOMBI
Affiliation:
Mathematical Institute, University of St Andrews, North Haugh, St Andrews, Fife KY16 9SS e-mail: edombi@gmail.com
N. D. GILBERT
Affiliation:
School of Mathematical and Computer Sciences, Heriot-Watt University, Riccarton, Edinburgh EH14 4AS e-mail: ndg@ma.hw.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

We study a construction of an HNN extension for inverse semigroups with zero. We prove a normal form for the elements of the universal group of an inverse semigroup that is categorical at zero, and use it to establish structural results for the universal group of an HNN extension. Our main application of the HNN construction is to show that graph inverse semigroups –including the polycyclic monoids –admit HNN decompositions in a natural way, and that this leads to concise presentations for them.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 142 , Issue 1 , January 2007 , pp. 25 - 39
Copyright
Copyright © Cambridge Philosophical Society 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1] Ash, C. J. and Hall, T. E.. Inverse semigroups on graphs. Semigroup Forum 11 (1975), 140–145.CrossRefGoogle Scholar
[2] Dombi, E. R. and Gilbert, N. D.. HNN constructions of free inverse semigroups and of Bruck-Reilly extensions. Preprint.Google Scholar
[3] E. R. Dombi, N. D. Gilbert and N.Ru kuc. Finite presentability of HNN extensions of inverse semigroups. Internat. J. Algebra Comput. 15 (2005), 423–436.Google Scholar
[4] Gilbert, N. D.. HNN extensions of inverse semigroups and groupoids. J. Algebra 272 (2004), 27–45.CrossRefGoogle Scholar
[5] Gomes, G. M. S. and Howie, J. M.. A P–theorem for inverse semigroups with zero. Portugaliae Mathematica 53 (1996), 257–278.Google Scholar
[6] Higgins, P. J.. Notes on categories and groupoids. Van Nostrand Reinhold Math. Stud. 32 (1971). Reprinted electronically at www.tac.mta.co/tac/reprints/articles/7/7tr7.pdf.Google Scholar
[7] Howie, J. M.. Fundamentals of Semigroup Theory. London Math. Soc. Monogr. (1997).Google Scholar
[8] Lawson, M. V.. Inverse Semigroups (World Scientific, 1998).CrossRefGoogle Scholar
[9] Lawson, M. V.. Constructing inverse semigroups from category actions. J. Pure Appl. Algebra 137 (1999), 57–101.CrossRefGoogle Scholar
[10] Lawson, M. V.. E*–unitary inverse semigroups. In Semigroups, Automata, Algebra and Languages (Gomes, G. M. S. et al. , Eds.) (World Scientific, 2002).Google Scholar
[11] Lawson, M. V.. Left cancellative categories and ordered groupoids. Semigroup Forum 68 (2004), 458–476.CrossRefGoogle Scholar
[12] Munn, W. D.. Uniform semilattices and bisimple inverse semigroups. Quar t. J. Math. Oxford (2) 17 (1966), 151–159.CrossRefGoogle Scholar
[13] Petrich, M.. Inverse Semigroups (John Wiley & Sons, 1984).Google Scholar
[14] Yamamura, A.. HNN extensions of inverse semigroups and applications. Internat. J. Algebra Comput. 7 (1997), 605–624.CrossRefGoogle Scholar
[15] Yamamura, A.. HNN extensions of semilattices. Internat. J. Algebra Comput. 9 (1999), 555–596.CrossRefGoogle Scholar