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Naturally ordered regular semigroups with a greatest idempotent

Published online by Cambridge University Press:  14 November 2011

T. S. Blyth  and
R. McFadden
T. S. Blyth
Affiliation:
Mathematical Institute, University of St Andrews
R. McFadden
Affiliation:
Department of Mathematical Sciences, Northern Illinois University, Dekalb, Illinois, U.S.A
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Synopsis

We consider ordered regular semigroups in which the order extends the natural order on the idempotents, and which are graced with the presence of a greatest idempotent. This implies that every element has a greatest inverse. An investigation into the properties ofthese special elements allows a description of Green's relations on the subsemigroup generated by the idempotents. This in turn leads to a complete description of the structure of idempotent-generated naturally ordered regular semigroups having a greatest idempotent. The smallest such semigroup that is not orthodox is also described. These results lead us to obtain structure theorems in the general case with the added condition that Green's relations be regular. Finally, necessary and sufficient conditions for such a semigroup to be a Dubreil-Jacotin semigroup are found.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 91 , Issue 1-2 , 1981 , pp. 107 - 122
Copyright
Copyright © Royal Society of Edinburgh 1981

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