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# 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.

## Mathematics Subject Classification

20M18*no label found*

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