Idempotent-separating extensions of regular semigroups. (Q2368413)

Summary: For a regular biordered set \(E\), the notion of \(E\)-diagram and the associated regular semigroup was introduced in our previous paper [\textit{M. Loganathan, A. Tamilarasi}, Semigroup Forum 51, No. 2, 191-216 (1995; Zbl 0833.20077)]. 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 \(\text{Reg}_E(\mathbf A)\) having \(E\) as its biordered set of idempotents. This regular semigroup is analogous to the automorphism group of a group. This paper provides an application of \(\text{Reg}_E(\mathbf A)\) 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.