Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-09T05:24:13.956Z Has data issue: false hasContentIssue false

Completely simple and inverse semigroups

Published online by Cambridge University Press:  24 October 2008

R. McFadden
Affiliation:
Queen's UniversityBelfast
Hans Schneider
Affiliation:
Queen's UniversityBelfast

Extract

The purpose of this paper is to investigate the structure of certain types of semigroups. Rees(6),(7) has determined the structure of a completely simple semigroup, and has shown that such a system may be realized as a type of matrix semigroup. Clifford (2) and Schwarz (8) have found conditions, namely, the existence of minimal left and minimal right ideals, under which a simple semigroup is completely simple, and have made a more detailed study of such semigroups. Preston (4), (5) has studied inverse semigroups, in which each non-zero element has a unique relative inverse, and has also considered inverse semigroups which contain minimal right or left ideals. In the present paper we obtain a set of conditions on a simple semigroup, each of which is equivalent to the semigroup being both completely simple and inverse. Section 2 defines the terms used and gives a brief resume of the main results which have already been proved. Section 3 is devoted to our present considerations.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1961

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Bruch, R. H., A survey of binary systems (Berlin, 1958).CrossRefGoogle Scholar
(2)Clifford, A. H., Semigroups without nilpotent ideals. Amer. J. Math. 71 (1949), 834–44.CrossRefGoogle Scholar
(3)Munn, W. D., and Penrose, R., A note on inverse semigroups. Proc. Camb. Phil. Soc. 51 (1955), 396–9.CrossRefGoogle Scholar
(4)Preston, G. B., Inverse semigroups. J. Lond. Math. Soc. 29 (1954), 397403.Google Scholar
(5)Preston, G. B., Inverse semigroups with minimal right ideals. J. Lond. Math. Soc. 29 (1954), 404–11.CrossRefGoogle Scholar
(6)Rees, D., On semigroups. Proc. Camb. Phil. Soc. 36 (1940), 387400.CrossRefGoogle Scholar
(7)Rees, D., A note on semigroups. Proc. Camb. Phil. Soc. 37 (1941), 434–5.Google Scholar
(8)Schwarz, S., On semigroups having a kernel. Czech. Math. J. 1 (76) (1951).CrossRefGoogle Scholar