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Completely simple and inverse semigroups

Published online by Cambridge University Press:  24 October 2008

R. McFadden  and
Hans Schneider
R. McFadden
Affiliation:
Queen's UniversityBelfast
Hans Schneider
Affiliation:
Queen's UniversityBelfast
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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
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 57 , Issue 2 , April 1961 , pp. 234 - 236
Copyright
Copyright © Cambridge Philosophical Society 1961

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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), 397–403.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), 387–400.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