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'Wagner-Preston representation theorem'
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| Title of object: |
Wagner-Preston representation theorem |
| Canonical Name: |
WagnerPrestonRepresentationTheorem |
| Type: |
Theorem |
| Created on: |
2006-08-21 16:39:52 |
| Modified on: |
2006-08-22 07:17:49 |
| Classification: |
msc:20M18 |
| Keywords: |
Inverse Semigroups |
| Defines: |
semigroup representation via bijective partial map, faithful representation, Wagner-Preston representation |
Preamble:
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Content:
\newcommand{\domi}{\mathrm{dom}}
\newcommand{\rang}{\mathrm{ran}}
\newcommand{\FFF}{\mathfrak{F}}
\newcommand{\III}{\mathfrak{I}}
\newcommand{\cbra}[1]{\left( #1 \right)}
\newcommand{\qbra}[1]{\left[ #1 \right]}
\newcommand{\gbra}[1]{\left\{ #1 \right\}}
\newcommand{\abra}[1]{\left\langle #1 \right\rangle}
Let $S$ be an inverse semigroup and $X$ a set. An inverse semigroup homomorphism $\phi:S\rightarrow\III(X)$ is called a \emph{representation} of $S$ by bijective partial maps on $X$. The representation is said to be \emph{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 \emph{Wagner-Preston representation} of $S$. The following result, due to Wagner and Preston, is analogous to the Cayley representation theorem for groups.\\
\begin{thm}[\textbf{Wagner-Preston representation theorem}]
The Wagner-Preston representation of an inverse semigroup is faithful.
\end{thm}
\begin{thebibliography}{9}
\bibitem{b:petrich} N. Petrich, \emph{Inverse Semigroups}, Wiley, New York, 1984.
\bibitem{b:pres} G.B. Preston, \emph{Representation of inverse semi-groups}, J. London Math. Soc. 29 (1954), 411-419.
\end{thebibliography} |
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