Wagner-Preston representation theorem

Let $S$ be an inverse semigroup and $X$ a set. An inverse semigroup homomorphism $\varphi :S\to \Im (X)$, where $\Im (X)$ denotes the symmetric inverse semigroup, is called a representation of $S$ by bijective partial maps on $X$. The representation is said to be faithful if $\varphi$ is a monomorphism, i.e. it is injective.

Given $s\in S$, we define ${\rho }_s\in \Im (S)$ as the bijective partial map with domain

$$\mathrm{dom}({\rho }_s)=S{s}^{-1}=\{t{s}^{-1}|t\in S\}$$

and defined by

$${\rho }_s(t)=ts,\forall t\in \mathrm{dom}({\rho }_s).$$

Then the map $s\mapsto {\rho }_s$ is a representation called the Wagner-Preston representation of $S$. The following result, due to Wagner and Preston, is analogous to the Cayley representation theorem for groups.

Theorem 1 (Wagner-Preston representation theorem)

The Wagner-Preston representation of an inverse semigroup is faithful.