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Wagner-Preston representation theorem

(Theorem)

Let be an inverse semigroup and 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 by bijective partial maps on . 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

$\displaystyle \mathrm{dom}(\rho_s)=Ss^{-1}=\left\{ ts^{-1}\,\vert\,t\in S \right\}$

and defined by

$\displaystyle \rho_s(t)=ts,\ \ \forall t\in \mathrm{dom}(\rho_s).$

Then the map $s\mapsto\rho_s$ is a representation called the Wagner-Preston representation of

. 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.

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.

"Wagner-Preston representation theorem" is owned by Mazzu.

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Also defines:

representation by bijective partial maps, faithful representation, Wagner-Preston representation

Keywords:

Inverse Semigroups

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20M18 (Group theory and generalizations :: Semigroups :: Inverse semigroups)

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