Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-23T04:55:47.943Z Has data issue: false hasContentIssue false
  • English
  • Français

Proper Dubreil-Jacotin inverse semigroups

Published online by Cambridge University Press:  18 May 2009

R. McFadden
R. McFadden
Affiliation:
Department of Mathematical Sciences, Northern Illinois University, De Kalb, Illinois 60115, U.S.A.
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.

This paper is concerned mainly with the structure of inverse semigroups which have a partial ordering defined on them in addition to their natural partial ordering. However, we include some results on partially ordered semigroups which are of interest in themselves. Some recent information [1, 2, 6, 7,11] has been obtained about the algebraic structure of partially ordered semigroups, and we add here to the list by showing in Section 1 that every regular integrally closed semigroup is an inverse semigroup. In fact it is a proper inverse semigroup [10], that is, one in which the idempotents form a complete class modulo the minimum group congruence, and the structure of these semigroups is explicitly known [5].

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 16 , Issue 1 , March 1975 , pp. 40 - 51
Copyright
Copyright © Glasgow Mathematical Journal Trust 1975

References

REFERENCES

1.Blyth, T. S., Dubreil-Jacotin inverse semigroups (submitted to Proc. Roy. Soc. Edinburgh).Google Scholar
2.Blyth, T. S., The structure of certain integrally closed semigroups, Semigroup Forum 8, 1974, 104–124.Google Scholar
3.Blyth, T. S. and Janowitz, M. F., Residuation Theory, International Series of Monographs in Pure and Applied Mathematics, Volume 102. Pergamon Press, 1972.Google Scholar
4.Clifford, A. H. and Preston, G. B., The algebraic theory of semigroups, American Math. Soc. Survey 7, Volumes I and II (Providence, R.I., 1961 and 1967).Google Scholar
5.McAIister, D. B., Groups, semilattices and inverse semigroups II, Trans. Amer. Math. Soc., 196 (1974), 350–370.Google Scholar
6.McAIister, D. B. and O'Carroll, L., On B-nomal semigroups, J. London Math. Soc. (2), 2 (1970), 679–688.CrossRefGoogle Scholar
7.McFadden, R., On the structure of certain A-nomal semigroups, J. London Math. Soc. (2), 6 (1973), 543–552.CrossRefGoogle Scholar
8.McFadden, R. and O'Carroll, L., F-inverse semigroups, Proc. London Math. Soc. (3), 22 (1971), 652–666.CrossRefGoogle Scholar
9.Munn, W. D., On fundamental inverse semigroups, Quart. J. Math. Oxford (2), 21 (1970), 157–170.CrossRefGoogle Scholar
10.Saito, T., Proper ordered inverse semigroups, Pacific J. Math. 15 (1965), 649–666.CrossRefGoogle Scholar
11.Saito, T., Ordered inverse semigroups, Trans. Amer. Math. Soc. 153 (1971), 99–138.CrossRefGoogle Scholar