Representations of inverse semigroups by one-to-one partial transformations of a set

We show that each representation ϕ, say, of an inverse semigroup S, by means of transformations of a set X, determines a representation ϕ^* by means of partial one-to-one transformations of X, in such a fashion that sϕ ↦ sϕ^*, for s ∈ S, is an isomorphism of Sϕ upon Sϕ^*. An immediate corollary is the classical faithful representation of an inverse semigroup as a semigroup of partial one-to-one transformations.