A biordered set representation of regular semigroups

In this paper, for an arbitrary regular biordered set E, by using biorder-isomorphisms between the omega-ideals of E, we construct a fundamental regular semigroup W-E called NH-semigroup of E, whose idempotent biordered set is isomorphic to E. We prove further that W-E can be used to give a new representation of general regular semigroups in the sense that, for any regular semigroup S with the idempotent biordered set isomorphic to E, there exists a homomorphism from S to W-E whose kernel is the greatest idempotent-separating congruence on S and the image is a full symmetric subsemigroup of W-E. Moreover, when E is a biordered set of a semilattice E-0, WE is isomorphic to the Munn-semigroup T-E0; and when E is the biordered set of a band B, WE is isomorphic to the Ball-semigroup W-B.