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Translational hull and semigroups of binary relations

Published online by Cambridge University Press:  18 May 2009

Mario Petrich
Mario Petrich
Affiliation:
The Pennsylvania State UniversityUniversity Park, Pennsylvania
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Various semigroups of partial transformations (and more generally, semigroups of binary relations) on a set have been studied by a number of Soviet mathematicians; to mention only a few: Gluskin [2], Ljapin [4], Shutov [6], Zaretski [7], [8]. In their study the densely embedded ideal of a semigroup introduced by Ljapin [4] plays a central role. In fact, a concrete semigrou Q is described in several instances by its abstract characteristic, namely either by a set of postulates on an abstract semigroup or by a set of postulates (which are usually much simpler) on an abstract semigroup S which is a densely embedded ideal of a semigroup T isomorphic to Q. In many cases, the densely embedded ideal S is a completely 0-simple semigroup. The following theorem [3, 1.7.1] reduces the study of a semigroup Q with a weakly reductive densely embedded ideal S to the study of the translational hull of S:

Theorem (Gluskin). If S is a weakly reductive densely embedded ideal of a semigroup Q, then Q is isomorphic to the translational hull ω(S) of S.

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 9 , Issue 1 , January 1968 , pp. 12 - 21
Copyright
Copyright © Glasgow Mathematical Journal Trust 1968

References

REFERENCES

1.Clifford, A. H. and Preston, G. B., The algebraic theory of semigroups, Vol. I, Math. Surveys No. 7, Amer. Math. Soc. (Providence, R. I., 1961).CrossRefGoogle Scholar
2.Gluskin, L. M., Ideals of transformation semigroups, Mat. Sb. 47 (1959), 111–130 (Russian).Google Scholar
3.Gluskin, L. M., Ideals of semigroups, Mat. Sb. 55 (1961), 421–448 (Russian).Google Scholar
4.Ljapin, E. S., Associative systems of all partial transformations, Doklady Akad. Nauk SSSR 88 (1953), 13–15 (errata 92 (1953), 692) (Russian).Google Scholar
5.Petrich, Mario, The translational hull of a completely 0-simple semigroup, Glasgow Math. J. 9 (1968), 1–11.CrossRefGoogle Scholar
6.Shutov, E. G., On semigroups of almost identical transformations, Doklady Akad. Nauk SSSR 134 (1960), 292–295 (Russian).Google Scholar
7.Zaretski, K. A., Abstract characteristic of the semigroup of all binary relations, Leningrad, gos. ped. inst. Uch. zap. 183 (1958), 251–262 (Russian).Google Scholar
8.Zaretski, K. A., Semigroup of binary relations, Mat. Sb. 61 (1963), 291–305 (Russian).Google Scholar