Hostname: page-component-745bb68f8f-v2bm5 Total loading time: 0 Render date: 2025-01-27T02:36:07.732Z Has data issue: false hasContentIssue false
  • English
  • Français

Baer-Levi semigroups of partial transformations

Published online by Cambridge University Press:  17 April 2009

Fernanda A. Pinto  and
R.P. Sullivan
Fernanda A. Pinto
Affiliation:
Centro de Matematica, Universidade do Minho, 4710 Braga, Portugal
R.P. Sullivan
Affiliation:
School of Mathematics & Statistics, University of Western Australia, Nedlands W.A. 6009, Australia
Rights & Permissions [Opens in a new window]

Extract

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.

Let X be an infinite set and suppose א0 ≤ q ≤ |X|. The Baer-Levi semigroup on X is the set of all injective ‘total’ transformations α: X → X such that |X\Xα| = q. It is known to be a right simple, right cancellative semigroup without idempotents, its automorphisms are “inner”, and some of its congruences are restrictions of Malcev congruences on I(X), the symmetric inverse semigroup on X. Here we consider algebraic properties of the semigroup consisting of all injective ‘partial’ transformations α of X such that |X\Xα| = q: in particular, we descried the ideals and Green's relations of it and some of its subsemigroups.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 69 , Issue 1 , February 2004 , pp. 87 - 106
Copyright
Copyright © Australian Mathematical Society 2004

References

[1] Clifford, A.H. and Preston, G.B., The algebraic theory of semigroups, Mathematical Surveys, No. 7, vol 1 and 2 (American Mathematical Society, Providence, RI, 1961 and 1967).Google Scholar
[2] Howie, J.M., An introduction to semigroup theory (Academic Press, London, 1976).Google Scholar
[3] Howie, J.M., ‘A congruence-free inverse semigroup associated with a pair of infinite cardinals’, J. Austral. Math. Soc. Ser. A 31 1981, 337–342.CrossRefGoogle Scholar
[4] Levi, I., ‘Automorphisms of normal partial transformation semigroups’, Glasgow Math. J. 29 1987, 149–157.CrossRefGoogle Scholar
[5] Levi, I., Schein, B.M., Sullivan, R.P. and Wood, G. R., ‘Automorphism of Baer-Levi semigroups’, J. London Math. Soc. 28 1983, 492–495.CrossRefGoogle Scholar
[6] Lindsey, D. and Madison, B., ‘The lattice of congruences on a Baer-Levi semigroup’, Semigroup Forum 12 1976, 63–70.CrossRefGoogle Scholar
[7] Marques-Smith, M.P.O. and Sullivan, R.P., ‘The ideal structure of nilpotent-generated transformation semigroups’, Bull. Austral. Math. Soc. 60 1999, 303–318.CrossRefGoogle Scholar
[8] Reynolds, M.A. and Sullivan, R.P., ‘The ideal structure of idempotent-generated transformation semigroups’, Proc Edinburgh Math. Soc. 28 1985, 319–331.CrossRefGoogle Scholar
[9] Sullivan, R.P., ‘Automorphisms of transformation semigroups’, J. Austral. Math. Soc. 20 1975, 77–84.CrossRefGoogle Scholar
[10] Sullivan, R.P., ‘Semigroups generated by nilpotent transformations’, J. Algebra 110 1987, 324–343.CrossRefGoogle Scholar