PresentationOfInverseMonoidsAndInverseSemigroups 2006-08-21 05:59:47 2006-08-24 12:00:05 Definition Inverse Semigroups Word Problem Isomorphism Problem % 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 % define commands here \newcommand{\e}{\mathrm{e}} \newcommand{\co}{\mathrm{c}} \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} \newcommand{\mipres}[2]{\mathrm{Inv}^1\abra{#1 | #2}} \newcommand{\sipres}[2]{\mathrm{Inv}\abra{#1 | #2}} \newcommand{\double}[1]{\cbra{#1\amalg #1^{-1}}} \newcommand{\doubles}[1]{\cbra{#1\amalg #1^{-1}}^\ast} \newcommand{\doublep}[1]{\cbra{#1\amalg #1^{-1}}^+} \newcommand{\fim}{\mathrm{FIM}} \newcommand{\fis}{\mathrm{FIS}} Let $\doubles{X}$ be the free monoid with involution on $X$, and $T\subseteq \doubles X\times \doubles X$ be a binary relation between words. We denote by $T^\e$ [resp. $T^\co$] the equivalence relation [resp. congruence] generated by $T$. A \emph{presentation (for an inverse monoid)} is a couple $(X;T)$. We use this couple of objects to define an inverse monoid $\mipres{X}{T}$. Let $\rho_X$ be the Wagner congruence on $X$, we define the inverse monoid $\mipres{X}{T}$ \emph{presented} by $(X;T)$ as $$\mipres{X}{T}=\doubles{X}/(T\cup\rho_X)^\co.$$ In the previous dicussion, if we replace everywhere $\doubles X$ with $\doublep X$ we obtain a \emph{presentation (for an inverse semigroup)} $(X;T)$ and an inverse semigroup $\sipres{X}{T}$ \emph{presented} by $(X;T)$. A trivial but important example is the Free Inverse Monoid [resp. Free Inverse Semigroup] on $X$, that is usually denoted by $\fim(X)$ [resp. $\fis(X)$] and is defined by $$\fim(X)=\mipres{X}{\varnothing}=\doubles{X}/\rho_X,\ \ \mbox{[resp. $\fis(X)=\sipres{X}{\varnothing}=\doublep{X}/\rho_X$]}.$$ \begin{thebibliography}{9} \bibitem{b:petrich} N. Petrich, \emph{Inverse Semigroups}, Wiley, New York, 1984. \bibitem{b:steph} J.B. Stephen, \emph{Presentation of inverse monoids}, J. Pure Appl. Algebra 63 (1990) 81-112. \end{thebibliography}