# Wagner-Preston representation theorem

Let $S$ be an inverse semigroup and $X$ a set. An inverse semigroup homomorphism $\phi:S\rightarrow\mathfrak{I}(X)$, where $\mathfrak{I}(X)$ 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 $\phi$ is a monomorphism, i.e. it is injective.

Given $s\in S$, we define $\rho_{s}\in\mathfrak{I}(S)$ as the bijective partial map with domain

 $\mathrm{dom}(\rho_{s})=Ss^{-1}=\left\{ts^{-1}\,|\,t\in S\right\}$

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.

## References

• 1 N. Petrich, Inverse Semigroups, Wiley, New York, 1984.
• 2 G.B. Preston, Representation of inverse semi-groups, J. London Math. Soc. 29 (1954), 411-419.
Title Wagner-Preston representation theorem WagnerPrestonRepresentationTheorem 2013-03-22 16:11:16 2013-03-22 16:11:16 Mazzu (14365) Mazzu (14365) 10 Mazzu (14365) Theorem msc 20M18 representation by bijective partial maps faithful representation Wagner-Preston representation