Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-28T16:51:36.436Z Has data issue: false hasContentIssue false
  • English
  • Français

The Archimedean property in an ordered semigroup

Published online by Cambridge University Press:  09 April 2009

Tôru Saitô
Tôru Saitô
Affiliation:
Tokyo Gakugei University
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.

By an ordered semigroup we mean a semigroup with a simple order which is compatible with the semigroup operation. Several authors, for example Alimov [1], Clifford [2], Conrad [4] and Hion [7], studied the archimedean property in some special kinds of ordered semigroups. For a general ordered semigroup, Fuchs [6] defined the archimedean equivalence as follows: a ~ b if and only if one of the four conditionsholds for some positive integer n.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 8 , Issue 3 , August 1968 , pp. 547 - 556
Copyright
Copyright © Australian Mathematical Society 1968

References

[1]Alimov, N. G., ‘On ordered semigroups’, Izv. Akad. Nauk SSSR 14 (1950), 569–576 (Russian).Google Scholar
[2]Clifford, A. H., ‘Naturally totally ordered commutative semigroups’, Amer. J. Math. 70 (1954), 631–646.CrossRefGoogle Scholar
[3]Clifford, A. H. and Preston, G. B., The algebraic theory of semigroups I (Amer. Math. Soc. Math. Surveys No. 7, 1961).Google Scholar
[4]Conrad, P., ‘Ordered semigroups’, Nagoya Math. J. 16 (1960), 51–64.CrossRefGoogle Scholar
[5]Dubreil-Jacotin, M. L., Lesieur, L. and Croisot, R., Leçons sur la théorie des treillis, des structures algébriques ordonnées et des treillis géométriques (Gauthier-Villars, 1953).Google Scholar
[6]Fuchs, L., Partially ordered algebraic systems (Pergamon Press, 1963).Google Scholar
[7]Hion, Ya. V., ‘Ordered semigroups’, Izv.Akad. Nauk SSSR 21 (1957), 209–222 (Russian).Google Scholar
[8]Saitô, T., ‘Ordered idempotent semigroups’, J. Math. Soc. Japan 14 (1962), 150–169.CrossRefGoogle Scholar
[9]Saitô, T., ‘Regular elements in an ordered semigroup’, Pacific J. Math. 13 (1963), 263–295. Correction, 14 (1964), 1505.CrossRefGoogle Scholar
[10]Gabovits, E. Ya., Ordered semigroups (Autoreview of dissertation, Leningrad, 1967) (Russian).Google Scholar