WagnerPrestonRepresentationTheorem 2006-08-21 16:39:52 2007-03-15 08:41:48 Theorem representation by bijective partial maps faithful representation Wagner-Preston representation Inverse Semigroups % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % THEOREM Environments -------------------------------------------------- \newtheorem{thm}{Theorem} % define commands here \PMlinkescapeword{representation} \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 $\phi:S\rightarrow\III(X)$, where $\III(X)$ denotes the symmetric inverse semigroup, 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}