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Completely right injective semigroups that are unions of groups†

Published online by Cambridge University Press:  18 May 2009

E. H. Feller and
R. L. Gantos
E. H. Feller
Affiliation:
University of Wisconsin, Milwaukee, U.S.A.
R. L. Gantos
Affiliation:
University of Wisconsin, Milwaukee, U.S.A.
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A semigroup S with 0 and 1 is termed completely right injective provided every right unitary S-system is injective. A necessary condition for a semigroup to be com-pletely right injective is given in [2]; namely, every right ideal is generated by an idempotent. An example in section 3 of this paper shows the existence of semigroups with 0 and 1 satisfying this condition which are not completely right injective. In [3], it is shown that the condition that every right and left ideal is generated by an idempotent is necessary and sufficient in the case that S is both completely right and left injective (called completely injective). Such a semigroup is an inverse semigroup with 0 whose idempotents are dually well-ordered.

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 12 , Issue 1 , April 1971, pp. 43 - 49
Copyright
Copyright © Glasgow Mathematical Journal Trust 1971

References

REFERENCES

1.Clifford, A. H. and Preston, G. B., The algebraic theory of semigroups, Amer. Math. Soc. Mathematical Surveys 7, Vol. I (Providence, R. I., 1961).Google Scholar
2.Feller, E. H. and Gantos, R. L., Completely injective semigroups with central idempotents, Glasgow Math. J. 10 (1969), 16–20.Google Scholar
3.Feller, E. H. and Gantos, R. L., Completely injective semigroups, Pacific J. Math. 31 (1969), 359–366.CrossRefGoogle Scholar