Fundamental Regular Semigroups with Quasi-ideal Regular *-Transversals

Let S be a semigroup and "*" a unary operation on S which satisfies the following identities xx* x = x, x* xx* = x*, x*** = x*, (xy*)* = y** x*, (x* y)* = y* x**. Then, S* = {x*|x is an element of S} is called a regular *-transversal of S in the literatures. Following Munn and Hall's idea, in this paper we construct fundamental regular semigroups with quasi-ideal regular *-transversals by which fundamental representations of regular semigroups with quasi-ideal regular *-transversals are obtained.