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
Canonical name	WagnerPrestonRepresentationTheorem
Date of creation	2013-03-22 16:11:16
Last modified on	2013-03-22 16:11:16
Owner	Mazzu (14365)
Last modified by	Mazzu (14365)
Numerical id	10
Author	Mazzu (14365)
Entry type	Theorem
Classification	msc 20M18
Defines	representation by bijective partial maps
Defines	faithful representation
Defines	Wagner-Preston representation