A Biordered Set Representation Of Regular Semigroups And Applications

Since the algebraic theory of semigroups was independent of other branches of algebra in 60's of 20th century, the study of properties and structures of various classes of regular semigroups has been the dominating aspect in the theory and resulted to a quite abundant and complete subject, the theory of regular semigroups. In the theory of regular semigroups, a constructive theorem, due to W.D.Munn, for inverse semigroups and a structural theorem, due to T.E.Hall, for orthodox semigroups established two important milestones in the development of the study. They gave concrete constructions of a fundamental inverse semigroup TE (the Munn's semigroup of the semilattice E) and a fundamental orthodox semigroup WB (the Hall's semigroup of the band B) in terms of isomorphisms between principal ideals in a semilattice E or band B respectively. On the basis of these two constructions, Munn and Hall gave fundamental representations for these two classes of regular semigroups. As for construction of regular semigroups in general, a famous India mathematician K.S.S.,Nambooripad initiated the theory of regular biordered sets and solved in global the structure of arbitrary regular semigroups by using equivalence of categories. In the first two sections of this thesis, by using isomorphisms between principal ideal in an arbitrary regular biordered set E, we give a concrete construction of a fundamental regular semigroup WE (named NH semigroup of biordered set E). On the basis of this construction we give a fundamental representation for arbitrary regular semigroups (see Theorem 2.21). In particular, when E is a semilattice or band biordered set, our WE is exactly isomorphic to the Munn's semigroup TE or the Hall's semigroup WB (E(B) = E) respectively. Thus our work generalizes both representation theories due to W.D.Munn and T.E.Hall for inverse and orthodox semigroups to general regular semigroups.The concept of weakly inverse semigroup was introduced by another Indian mathematician B.R.,Srinivasan in 1968 as a generalization of inverse semigroups. He proved that the set PT(A) of all partial transformations on an arbitrary set A forms a weakly inverse semigroup, called the symmetric weakly inverse semigroup, and that any weakly inverse semigroup S can be embedded into the symmetric weakly inverse semigroup PT(S). In 1976, K.S.S., Nambooripad generalized this concept to that of (LR) semigroup and gave a sufficient and necessary conditions for an (LR) semigroup to be weakly inverse semigroup. In 1999, YU Shi-wei etc., proved that any idempotent-separating homomorphic image of the symmetry weakly inverse semigroup is still weakly inverse. But the problem whether or not idempotent-separating homomorphic images of an arbitrary weakly inverse semigroup are also weakly inverse is left open. In the third section of this thesis, we will give sufficient and necessary conditions for an idempotent-separating homomorphic image of any weakly inverse semigroup to be weakly inverse. Inparticular, we will describe sufficient and necessary conditions for the NH semigroup WE to be weakly inverse and whence we find out a new particular class of regular biordered sets ?fundamental weakly biordered sets. Moreover, by using of the construction of WE, we will give a counterexample to show that there exists a weakly inverse semigroup one of whose idempotent-separating homomorphic image is not a weakly inverse. Thus the open problem mentioned above, is completely solved.