Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2025-01-03T18:43:54.892Z Has data issue: false hasContentIssue false
  • English
  • Français

Inverse semigroups and their natural order

Published online by Cambridge University Press:  17 April 2009

H. Mitsch
H. Mitsch
Affiliation:
Mathematisches Institut der Universität Wien, Wien, Austria.
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.

The natural order of an inverse semigroup defined by a ≤ b ⇔ a′b = a′a has turned out to be of great importance in describing the structure of it. In this paper an order-theoretical point of view is adopted to characterise inverse semigroups. A complete description is given according to the type of partial order an arbitrary inverse semigroup S can possibly admit: a least element of (S, ≤) is shown to be the zero of (S, ·); the existence of a greatest element is equivalent to the fact, that (S, ·) is a semilattice; (S, ≤) is directed downwards, if and only if S admits only the trivial group-homomorphic image; (S, ≤) is totally ordered, if and only if for all a, b ∈ S, either ab = ba = a or ab = ba = b; a finite inverse semigroup is a lattice, if and only if it admits a greatest element. Finally formulas concerning the inverse of a supremum or an infimum, if it exists, are derived, and right-distributivity and left-distributivity of multiplication with respect to union and intersection are shown to be equivalent.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 19 , Issue 1 , August 1978 , pp. 59 - 65
Copyright
Copyright © Australian Mathematical Society 1979

References

[1]Birkhoff, Garrett, Lattice theory, Third edition (American Mathematical Society Colloquium Publications, 25. American Mathematical Society, Providence, Rhode Island, 1967).Google Scholar
[2]Clifford, A.H. and Preston, G.B., The algebraic theory of semigroups, Volume I (Mathematical Surveys, 7. American Mathematical Society, Providence, Rhode Island, 1961; reprinted 1964).Google Scholar
[3]Clifford, A.H. and Preston, G.B., The algebraic theory of semigroups, Volume II (Mathematical Surveys, 7. American Mathematical Society, Providence, Rhode Island, 1967; reprinted 1971).Google Scholar
[4]Dubreil, Paul, Contribution à la théorie des semi-groupes (Mém. Acad. Sci. Inst. France (2) 63, no. 3, 52 pp (1941)).Google Scholar
[5]Fuchs, L., Partially ordered algebraic systems (Pure and Applied Mathematics, 28. Akadémiai Kiadó, Budapest; Pergamon Press, Oxford, London, New York, Paris, 1963).Google Scholar
[6]Howie, J.M., An introduction to semigroup theory (London Mathematical Society Monographs, 7. Academic Press [Harcourt Brace Jovanovich], London, New York, San Francisco, 1976).Google Scholar
[7]Munn, W.D., “A class of irreducible matrix representations of an arbitrary inverse semigroup”, Proc. Glasgow Math. Assoc. 5 (1961–1962), 41–48.CrossRefGoogle Scholar
[8]Вагнер, В.В. [V.V. Vagner], “Обобщенные грУппы” [Generalized groups], Dokl. Akad. Nauk SSSR (N.S.) 84 (1952), 1119–1122.Google ScholarPubMed