Hostname: page-component-8448b6f56d-sxzjt Total loading time: 0 Render date: 2024-04-24T04:50:57.446Z Has data issue: false hasContentIssue false
  • English
  • Français

EMBEDDINGS IN COSET MONOIDS

Part of: Semigroups

Published online by Cambridge University Press:  01 August 2008

JAMES EAST
JAMES EAST*
Affiliation:
School of Mathematics and Statistics, Carslaw Building F07, University of Sydney, New South Wales 2006, Australia (email: jamese@maths.usyd.edu.au)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A submonoid S of a monoid M is said to be cofull if it contains the group of units of M. We extract from the work of Easdown, East and FitzGerald (2002) a sufficient condition for a monoid to embed as a cofull submonoid of the coset monoid of its group of units, and show further that this condition is necessary. This yields a simple description of the class of finite monoids which embed in the coset monoids of their group of units. We apply our results to give a simple proof of the result of McAlister [D. B. McAlister, ‘Embedding inverse semigroups in coset semigroups’, Semigroup Forum20 (1980), 255–267] which states that the symmetric inverse semigroup on a finite set X does not embed in the coset monoid of the symmetric group on X. We also explore examples, which are necessarily infinite, of embeddings whose images are not cofull.

Keywords

factorizable inverse monoidcoset monoidsymmetric inverse semigroup

MSC classification

Secondary: 20M18: Inverse semigroups 20M30: Representation of semigroups; actions of semigroups on sets
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 85 , Issue 1 , August 2008 , pp. 75 - 80
Copyright
Copyright © 2008 Australian Mathematical Society

References

[1]Artin, E., ‘Theory of braids’, Ann. of Math. 48(2) (1947), 101–126.CrossRefGoogle Scholar
[2]Chen, S. Y. and Hsieh, S. C., ‘Factorizable inverse semigroups’, Semigroup Forum 8(4) (1974), 283–297.CrossRefGoogle Scholar
[3]Crisp, J. and Paris, L., ‘The solution to a conjecture of Tits on the subgroup generated by the squares of the generators of an Artin group’, Invent. Math. 145(1) (2001), 19–36.CrossRefGoogle Scholar
[4]Dombi, E., ‘Almost factorizable straight locally inverse semigroups’, Acta Sci. Math. (Szeged) 69(3–4) (2003), 569–589.Google Scholar
[5]Easdown, D., East, J. and FitzGerald, D. G., ‘Braids and factorizable inverse monoids’, in: Semigroups and Languages (eds. I. M. Araújo, M. J. J. Branco, V. H. Fernandes and G. M. S. Gomes) (World Scientific, Singapore, 2002), pp. 86–105.Google Scholar
[6]Easdown, D., East, J. and FitzGerald, D. G., ‘Presentations of factorizable inverse monoids’, Acta Sci. Math. (Szeged) 71 (2005), 509–520.Google Scholar
[7]East, J., ‘Dual reflection monoids’, in preparation.Google Scholar
[8]East, J., ‘Factorizable inverse monoids of cosets of subgroups of a group’, Comm. Algebra 34 (2006), 2659–2665.CrossRefGoogle Scholar
[9]FitzGerald, D. G. and Leech, J., ‘Dual symmetric inverse semigroups and representation theory’, J. Aust. Math. Soc. 64 (1998), 345–367.Google Scholar
[10]Lawson, M. V., Inverse Semigroups. The Theory of Partial Symmetries (World Scientific, River Edge, NJ, 1998).CrossRefGoogle Scholar
[11]McAlister, D. B., ‘Embedding inverse semigroups in coset semigroups’, Semigroup Forum 20 (1980), 255–267.CrossRefGoogle Scholar
[12]Mills, J. E., ‘Combinatorially factorizable inverse monoids’, Semigroup Forum 59(2) (1999), 220–232.CrossRefGoogle Scholar
[13]Munn, W. D., ‘Uniform semilattices and bisimple inverse semigroups’, Quart. J. Math. Oxford (2) 17 (1966), 151–159.CrossRefGoogle Scholar
[14]Schein, B. M., ‘Semigroups of strong subsets’, Volž. Mat. Sb. Vyp. 4 (1966), 180–186.Google Scholar
[15]Schein, B. M., ‘Cosets in groups and semigroups’, in: Proc. Conf. on Semigroups with Applications, Oberwolfach, 21–28 July 1991 (eds. J. M. Howie, W. D. Munn and H. J. Weinert) (World Scientific, Singapore, 1992), pp. 205–221.Google Scholar
[16]Tirasupa, Y., ‘Factorizable transformation semigroups’, Semigroup Forum 18(1) (1979), 15–19.CrossRefGoogle Scholar
[17]Tirasupa, Y., ‘Weakly factorizable inverse semigroups’, Semigroup Forum 18(4) (1979), 283–291.CrossRefGoogle Scholar