Let $S$ be an inverse semigroup and $X$ a set. An inverse semigroup homomorphism $\phi:S\rightarrow\III(X)$ , where $\III(X)$ 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 $\rho_s\in\III(S)$ as the bijective partial map with domain $$\domi(\rho_s)=Ss^{-1}=\gbra{ts^{-1}\,|\,t\in S}$$ and defined by $$\rho_s(t)=ts,\ \ \forall t\in \domi(\rho_s).$$ 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.

Bibliography

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.