In [3] and [5], the authors gave a representation of a generalized inverse *-semigroup S, which is a generalization of the Preston-Vagner representation. If S is fundamental, we can obtain a more precise representation of S. The purpose of this paper is to give a generalization of the Munn representation (see [6]) for fundamental generalized inverse *-semigroups. This paper is the improvement of our earlier announcement [4].
By introducing a new concept of a strong π-groupoid X(π; Y; {φe,f}), we shall construct a fundamental generalized inverse *-semigroup Tx(π)(M). Also, we shall show that a generalized inverse *-semigroup is fundamental if and only if it is *-isomorphic to a P-full generalized inverse *-subsemigroup of Tx(π)(M) on a strong π-groupoid X(π; Y; {φe,f}).