A Biordered Set Representation of Regular Semigroups

In this paper, for an arbitrary regular biordered set E, by using biorderisomorphisms between the ideals of E, we construct a fundamental regular semigroup W E called NHsemigroup 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 idempotentseparating 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, W E is isomorphic to the Munnsemigroup TE0; and when E is the biordered set of a band B, W E is isomorphic to the Hallsemigroup W B .