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International Journal of Mathematics and Mathematical Sciences
Volume 2005, Issue 18, Pages 2945-2975

Idempotent-separating extensions of regular semigroups

Department of Mathematics, Kongu Engineering College, Perundurai 638052, Tamil Nadu, India

Received 1 July 2004; Revised 10 March 2005

Copyright © 2005 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


For a regular biordered set E, the notion of E-diagram and the associated regular semigroup was introduced in our previous paper (1995). Given a regular biordered set E, an E-diagram in a category C is a collection of objects, indexed by the elements of E and morphisms of C satisfying certain compatibility conditions. With such an E-diagram A we associate a regular semigroup RegE(A) having E as its biordered set of idempotents. This regular semigroup is analogous to automorphism group of a group. This paper provides an application of RegE(A) to the idempotent-separating extensions of regular semigroups. We introduced the concept of crossed pair and used it to describe all extensions of a regular semigroup S by a group E-diagram A. In this paper, the necessary and sufficient condition for the existence of an extension of S by A is provided. Also we study cohomology and obstruction theories and find a relationship with extension theory for regular semigroups.