A Biordered Set Representation of Regular Semigroups

In this paper, for an arbitrary regular biordered set E, by using biorder–isomorphisms between the ω–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, W _E is isomorphic to the Munn–semigroup ; and when E is the biordered set of a band B, W _E is isomorphic to the Hall–semigroup W _B . This project is supported by the Key Research Foundations No. [1999]127 and No. [2002]48 of the Education Department of Sichuan Province and the Fundamental Application Research No. [01GY051-64] of the Department of Science and Technology of Sichuan Province