Idempotent-separating extensions of regular semigroups

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.