Abstract. It is well-known (see [4], [5]) that the set ${\cal I}_X$ of all partial one-to-one mappings on a set $X$ is an inverse semigroup, which is called {\it the symmetric inverse semigroup} on $X$, and that any inverse semigroup can be embedded up to isomorphism in ${\cal I}_X$ on a set $X$ (Preston-Vagner Representation Theorem). The purpose of this paper is to obtain a generalization of the Preston-Vagner representation for generalized inverse $*$-semigroups. Let ${\cal G}{\cal I}_{X(\pi ';\{\sigma_{e,f} \} )}$ be the set of all partial one-to-one $\pi $-mappings on a $\pi $-set $X(\pi ';\{\sigma_{e,f} \} )$ and ${\cal M}$ the structure sandwich set determined by $X(\pi ';\{\sigma_{e,f} \} )$. Then we shall show that ${\cal G}{\cal I}_{X(\pi ';\{\sigma_{e,f} \} )}({\cal M})$ is a generalized inverse $*$-semigroup, which is called {\it the} $\pi $-{\it symmetric generalized inverse $*$-semigroup on a} $\pi $-{\it set} $X(\pi ';\{\sigma_{e,f} \} )$ {\it with a structure sandwich set} ${\cal M}$, and that any generalized inverse $*$-semigroup can be embedded up to $*$-isomorphism in ${\cal G}{\cal I}_{X(\pi ';\{\sigma_{e,f} \} )}({\cal M})$ on a $\pi $-set $X(\pi ';\{\sigma_{e,f} \} )$.

AMS Subject Classification (1991): 20M20, 20M19, 20M17

Received January 26, 1995. (Registered under 5678/2009.)