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The structure mappings on a regular semigroup

Published online by Cambridge University Press:  20 January 2009

John Meakin
John Meakin
Affiliation:
University of Nebraska Lincoln, Nebraska 68588
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In (5) the author showed how to construct all inverse semigroups from their trace and semilattice of idempotents: the construction is by means of a family of mappings between ℛ-classes of the semigroup which we refer to as the structure mappings of the semigroup. In (7) (see also (8) and (9)) K. S. S. Nambooripad has adopted a similar approach to the structure of regular semigroups: he shows how to construct regular semigroups from their trace and biordered set of idempotents by means of a family of mappings between ℛ-classes and between ℒ-classes of the semigroup which we again refer to as the structure mappings of the semigroup. In the present paper we aim to provide a simpler set of axioms characterising the structure mappings on a regular semigroup than the axioms (R1)-(R7) of Nambooripad (9). Two major differences occur between Nambooripad's approach (9) and the approach adopted here: first, we consider the set of idempotents of our semigroups to be equipped with a partial regular band structure (in the sense of Clifford (3)) rather than a biorder structure, and second, we shall enlarge the set of structure mappings used by Nambooripad.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 21 , Issue 2 , September 1978 , pp. 135 - 142
Copyright
Copyright © Edinburgh Mathematical Society 1978

References

REFERENCES

(1) Baird, G. R., On semigroups and uniform partial bands, Semigroup Forum 4 (1972), 185–188.CrossRefGoogle Scholar
(2) Clifford, A. H., The fundamental representation of a regular semigroup, Math. Dept. Tulane Univ. (07, 1974); announced in Semigroup Forum 10 (1975), 84–92.Google Scholar
(3) Clifford, A. H., The partial groupoid of idempotents of a regular semigroup, Math. Dept. Tulane Univ. (09. 1974); announced in Semigroup Forum 10 (1975), 262–268.Google Scholar
(4) Clifford, A. H. and Preston, G. B., The algebraic theory of semigroups, Vol. 1, (Math. Surveys No. 7, Amer. Math. Soc, Providence, R. I. 1961).Google Scholar
(5) Meakin, John, On the structure of inverse semigroups, Semigroup Forum 12 (1976), 6–14.CrossRefGoogle Scholar
(6) Meakin, John, Coextensions of inverse semigroups, J. Algebra 46 (1977), 315–333.CrossRefGoogle Scholar
(7) Nambooripad, K. S. S., Structure of regular semigroups, (Thesis, University of Kerala, 09. 1973).Google Scholar
(8) Nambooripad, K. S. S., Structure of regular semigroups I: fundamental regular semigroups, Semigroup Forum 9 (1976), 354–363.CrossRefGoogle Scholar
(9) Nambooripad, K. S. S., Structure of regular semigroups II: the general case, Semigroup Forum 9 (1975), 364–371.CrossRefGoogle Scholar