Semidirect products of regular semigroups

Authors:
Peter R. Jones and Peter G. Trotter

Journal:
Trans. Amer. Math. Soc. **349** (1997), 4265-4310

MSC (1991):
Primary 20M17, 20M07

DOI:
https://doi.org/10.1090/S0002-9947-97-01638-3

MathSciNet review:
1355299

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Abstract | References | Similar Articles | Additional Information

Abstract: Within the usual semidirect product of regular semigroups and lies the set of its regular elements. Whenever or is completely simple, is a (regular) subsemigroup. It is this `product' that is the theme of the paper. It is best studied within the framework of existence (or e-) varieties of regular semigroups. Given two such classes, and , the e-variety generated by is well defined if and only if either or is contained within the e-variety of completely simple semigroups. General properties of this product, together with decompositions of many important e-varieties, are obtained. For instance, as special cases of general results the e-variety of locally inverse semigroups is decomposed as , where is the variety of inverse semigroups and is that of right zero semigroups; and the e-variety of -solid semigroups is decomposed as , where is the variety of completely regular semigroups and is the variety of groups. In the second half of the paper, a general construction is given for the e-free semigroups (the analogues of free semigroups in this context) in a wide class of semidirect products of the above type, as a semidirect product of e-free semigroups from and , ``cut down to regular generators''. Included as special cases are the e-free semigroups in almost all the known important e-varieties, together with a host of new instances. For example, the e-free locally inverse semigroups, -solid semigroups, orthodox semigroups and inverse semigroups are included, as are the e-free semigroups in such sub-e-varieties as strict regular semigroups, -solid semigroups for which the subgroups of its self-conjugate core lie in some given group variety, and certain important varieties of completely regular semigroups. Graphical techniques play an important role, both in obtaining decompositions and in refining the descriptions of the e-free semigroups in some e-varieties. Similar techniques are also applied to describe the e-free semigroups in a different `semidirect' product of e-varieties, recently introduced by Auinger and Polák. The two products are then compared.

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Additional Information

**Peter R. Jones**

Affiliation:
Department of Mathematics, Statistics and Computer Science, Marquette University, P.O. Box 1881, Milwaukee, Wisconsin 53201-1881

Email:
jones@mscs.mu.edu

**Peter G. Trotter**

Affiliation:
Department of Mathematics, University of Tasmania, Hobart, Australia, 7001

Email:
trotter@hilbert.maths.utas.edu.au

DOI:
https://doi.org/10.1090/S0002-9947-97-01638-3

Received by editor(s):
August 15, 1994

Additional Notes:
The authors are indebted to the Australian Research Council for their support of this research. The first author also gratefully acknowledges the support of National Science Foundation grant INT-8913404.

Article copyright:
© Copyright 1997
American Mathematical Society