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Construction of regular semigroups with inverse transversals

Published online by Cambridge University Press:  20 January 2009

Tatsuhiko Saito
Tatsuhiko Saito
Affiliation:
Department of General EducationShimonoseki University of FisheriesYoshimi, Shimonoseki 759–65, Japan
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Let S be a regular semigroup. An inverse subsemigroup S° of S is an inverse transversal if |V(x)∩S°| = 1 for each x∈S, where V(x) denotes the set of inverses of x. In this case, the unique element of V(x)∩S° is denoted by x°, and x°° denotes (x°)–1. Throughout this paper S denotes a regular semigroup with an inverse transversal S°, and E(S°) = E° denotes the semilattice of idempotents of S°. The sets {e∈S:ee° = e} and {f∈S:f°f=f} are denoted by Is and Λs, respectively, or simply I and Λ. Though each element of these sets is idempotent, they are not necessarily sub-bands of S. When both I and Λ are sub-bands of S, S° is called an S-inverse transversal. An inverse transversal S° is multiplicative if x°xyy°∈E°, and S° is weakly multiplicative if (x°xyy°)°∈E° for every x, y∈S. A band B is left [resp. right] regular if e f e = e f [resp. e f e = f e], and B is left [resp. right] normal if e f g = e g f [resp. e f g = f e g] for every e, f, g∈B. A subset Q of S is a quasi-ideal of S if QSQ ⊆ S.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 32 , Issue 1 , February 1989 , pp. 41 - 51
Copyright
Copyright © Edinburgh Mathematical Society 1989

References

REFERENCES

1.Blyth, T. S. and McFadden, R., Regular semigroups with a multiplicative inverse transversal, Proc. Roy. Soc. Edinburgh 92A (1982), 253–270.CrossRefGoogle Scholar
2.McAuster, D. B. and Blyth, T. S., Split orthodox semigroups, J. Algebra 51, (1978), 491–525.CrossRefGoogle Scholar
3.McAuster, D. B. and McFadden, R., Regular semigroups with inverse transversals, Quart. J. Math. Oxford 34 (1983), 459–474.CrossRefGoogle Scholar
4.McAlister, D. B. and McFadden, R., Semigroups with inverse transversals as matrix semigroups, Quart. J. Math. Oxford 35 (1984), 455–474.CrossRefGoogle Scholar
5.Saito, Tatsuhiko, Construction of a class of regular semigroups with an inverse transversal, to appear.Google Scholar
6.Saito, Tatsuhiko, Regular semigroups with a weakly multiplicative inverse transversal. Proc. 8th Symposium on Semigroups, Shimane Univ. (1985), 22–25.Google Scholar
7.Yamada, M., Introduction to Semigroup Theory (Maki-Shoten, Tokyo, 1979), in Japanese.Google Scholar
8.Yoshida, R., Left inverse semigroups with inverse transversals, preprint.Google Scholar
9.Yoshida, R., Regular semigroups with multiplicative inverse transversals II, preprint.Google Scholar