Hostname: page-component-848d4c4894-v5vhk Total loading time: 0 Render date: 2024-06-22T00:57:22.888Z Has data issue: false hasContentIssue false
  • English
  • Français

The Maximum Idempotent-Separating Congruence on an Inverse Semigroup

Published online by Cambridge University Press:  20 January 2009

J. M. Howie
J. M. Howie
Affiliation:
Department of Mathematics, The University, Glasgow, W.2
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.

A congruence ρ on a semigroup will be called idempotent-separating if each ρ-class contains at most one idempotent. It is shown below that there exists a maximum such congruence µ on every inverse semigroup S. Two characterisations of µ are found, and it is shown (a) that S/µ⋍E, the semilattice of idempotents of S, if and only if E is contained in the centre of S; (b) that µ is the identical congruence on S if and only if E is self-centralising, in a sense explained below.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 14 , Issue 1 , June 1964 , pp. 71 - 79
Copyright
Copyright © Edinburgh Mathematical Society 1964

References

REFERENCE

(1) Clifford, A. H., Semigroups admitting relative inverses, Annals of Math., 42 (1941), 1037–1049.CrossRefGoogle Scholar
(2) Clifford, A. H. and Preston, G. B., The Algebraic Theory of Semigroups, vol. 1, Math. Surveys of the American Math. Soc. 7 (Providence, R.I., 1961).Google Scholar
(3) Green, J. A., On the structure of semigroups, Annals of Math., 54 (1951), 163–172.CrossRefGoogle Scholar
(4) Munn, W. D., A class of irreducible matrix representations of an arbitrary inverse semigroup, Proc. Glasgow Math. Assoc., 5 (1961), 41–48.CrossRefGoogle Scholar
(5) Munn, W. D., A certain sublattice of the lattice of congruences on a regular semigroup, Proc. Cambridge Phil. Soc. (to appear).Google Scholar
(6) Preston, G. B., Inverse semi-groups, J. London Math. Soc., 29 (1954), 396–403.CrossRefGoogle Scholar
(7) Preston, G. B., Representations of inverse semi-groups, J. London Math. Soc., 29 (1954), 411–419.CrossRefGoogle Scholar
(8) Preston, G. B., A note on representations of inverse semigroups, Proc. American Math. Soc., 8 (1957), 1144–1147.CrossRefGoogle Scholar
(9) Šaǐn, B. M., Representations of generalised groups, Izvestiya Vysšikh Učebnykh Zavedeniǐ (Matematika), 28 (1962), 164–176 (Russian).Google Scholar
(10) Vagner, V. V., The theory of generalised groups and generalised heaps, Mat. Sbornik (N.S.), 32 (1953), 545–632 (Russian).Google Scholar
(11) Vagner, V. V., Generalised groups, Doklady Akad. Nauk SSSR, 84 (1952), 1119–1122 (Russian).Google Scholar