Geodesic
Completely right injective semigroups that are unions of groups†
Glasgow mathematical journal, Tome 12 (1971) no. 1, pp. 43-49

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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. 
Feller, E. H.; Gantos, R. L. Completely right injective semigroups that are unions of groups†. Glasgow mathematical journal, Tome 12 (1971) no. 1, pp. 43-49. doi: 10.1017/S0017089500001130
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[1] 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] 2.Feller, E. H. and Gantos, R. L., Completely injective semigroups with central idempotents, Glasgow Math. J. 10 (1969), 16–20. Google Scholar

[3] 3.Feller, E. H. and Gantos, R. L., Completely injective semigroups, Pacific J. Math. 31 (1969), 359–366. Google Scholar | DOI

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