A Representation for $F$-Regular Semigroups |
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Authors: | Bernd Billhardt |
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Affiliation: | (1) Universitat Kassel, FB-17 Mathematik/Informatik Hollandische Str. 36, D-34127 Kassel, Germany |
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Abstract: | A regular (inverse) semigroup S is called F-regular (F-inverse), if each class of the least group congruence S contains a greatest element with respect to the natural partial order on S. Such a semigroup is necessarily an E-unitary regular (hence orthodox) monoid. We show that each F-regular semigroup S is isomorphic to a well determined subsemigroup of a semidirect product of a band X by S/S, where X belongs to the band variety, generated by the band of idempotents ES of S. Our main result, Theorem 4, is the regular version of the corresponding fact for inverse semigroups, and might be useful to generalize further features of the theory of F-inverse semigroups to the F-regular case. |
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