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Semigroups satisfying minimal conditions

Published online by Cambridge University Press:  18 May 2009

W. D. Munn
W. D. Munn
Affiliation:
The University Glasgow
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In his fundamental paper, “On the structure of semigroups” [6], J. A. Green has examined certain important minimal conditions which may be satisfied bya semigroup S.We say that S satisfies the minimal condition on principal left ideals if every set of principal left ideals of S contains a minimal member with respect to inclusion:this condition is denoted by ℳ1. The corresponding conditions on principal rightideals and principal two-sided ideals are denoted by ℳr and ℳ1 respectively. The purpose of the present paper is to give some further results concerning these three conditions.Extensive use is made of the work of A. H. Clifford ([3] and [4]) onminimal ideals.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 3 , Issue 3 , December 1957 , pp. 145 - 152
Copyright
Copyright © Glasgow Mathematical Journal Trust 1957

References

REFERENCES

1.Baer, R. and Levi, F., Vollständige irreduzibele Systeme von Gruppenaxiomen, S.-B.Heidelberger Akad. Wiss. Bëitrdge zur Algebra,18 (1932), 1–12.Google Scholar
2.Clifford, A. H., Semigroups admitting relative inverses, Ann. of Math. (2), 42 (1941), 1037–1049.CrossRefGoogle Scholar
3.Clifford, A. H., Semigroups containing minimal ideals, Amer. J. Math., 70 (1948), 521–526.CrossRefGoogle Scholar
4.Clifford, A. H., Semigroups without nilpotent ideals, Amer. J. Math., 71 (1949), 834–844.CrossRefGoogle Scholar
5.Croisot, R., Demi-groupes inversifs et demi-groupes réunions de demi-groupes simples, Ann. Sci. École Norm. Sup. (3), 70 (1953), 361–379.CrossRefGoogle Scholar
6.Green, J. A., On the structure of semigroups, Ann. of Math. (2), 54 (1951), 163–172.CrossRefGoogle Scholar
7.Munn, W. D., On semigroup algebras, Proc. Cambridge Phil. Soc., 51 (1955), 1–15.CrossRefGoogle Scholar
8.Rees, D., On semi-groups, Proc. Cambridge Phil. Soc., 36 (1940), 387–400.CrossRefGoogle Scholar
9.Rees, D., Note on semi-groups, Proc. Cambridge Phil. Soc., 37 (1941), 434–435.CrossRefGoogle Scholar
10.Šchwarz, S., Zur Theorie der Halbgruppen, Sbornik Prdc. Prlrodovedeckej Faculty Slovenskej Vniverzity v Bralislave, 6 (1943), 64 pp. [Math. Reviews 10 (1949), 12.]Google Scholar
11.Šehwarz, S., On the structure of simple semigroups without zero,Czechoslovak. Math. J., 1 (76) (1951), 41–53. [Math. Reviews, 14 (1953), 12.]CrossRefGoogle Scholar