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Epis are onto for finite regular semigroups

Published online by Cambridge University Press:  20 January 2009

T. E. Hall  and
P. R. Jones
T. E. Hall
Affiliation:
Monash University, Clayton 3168, Australia
P. R. Jones
Affiliation:
Marquette University, Milwaukee, WI 53233, U. S. A.
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After preliminary results and definitions in Section 1, we show in Section 2 that any finite regular semigroup is saturated, in the sense of Howie and Isbell [8] (that is, the dominion of a finite regular semigroup U in a strictly containing semigroup S is never S). This is equivalent of course to showing that in the category of semigroups any epi from a finite regular semigroup is in fact onto. Note for inverse semigroups the stronger result, that any inverse semigroup is absolutely closed [11, Theorem VII. 2.14] or [8, Theorem 2.3]. Further, any inverse semigroup is in fact an amalgamation base in the class of semigroups [10], in the sense of [5]. These stronger results are known to be false for finite regular semigroups [8, Theorem 2.9] and [5, Theorem 25]. Whether or not every regular semigroup is saturated is an open problem.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 26 , Issue 2 , June 1983 , pp. 151 - 162
Copyright
Copyright © Edinburgh Mathematical Society 1983

References

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