Fundamental regular semigroups with inverse transversals. (Q1402905)

This paper deals with structure theory of fundamental regular semigroups; it follows T.~E.~Hall's line and establishes an analogous theory in the context of regular semigroups with an inverse transversal. Let \(C\) be an idempotent generated regular semigroup with inverse transversal \(C^\circ\). A fundamental regular semigroup \(T_{C,C^\circ}\) with inverse transversal is constructed such that the following holds. For each regular semigroup \(S\) with an inverse transversal \(S^\circ\) such that \(\langle E(S)\rangle=C\) and \(C\cap S^\circ=C^\circ\) there exists a homomorphism \(\phi\colon S\to T_{C,C^\circ}\) such that the kernel of \(\phi\) is the greatest idempotent separating congruence on \(S\) and \(\phi\) maps \(S^\circ\) into the inverse transversal \(T^\circ_{C,C^\circ}\) of \(T_{C,C^\circ}\). In particular, \(S\) is fundamental if and only if \(S\) is isomorphic to a full subsemigroup of \(T_{C,C^\circ}\) (that is, \(\phi\) is a monomorphism). If \(C\) happens to be fundamental then every fundamental regular semigroup \(S\) with inverse transversal \(S^\circ\) such that \(\langle E(S)\rangle=C\) and \(C^\circ=C\cap S^\circ\) is isomorphic to a full subsemigroup of \(T_{C,C^\circ}\). The construction is very much reminiscent of Hall's construction but is, due to the additional algebraic structure, a little bit less complicated.