|
|
|
Viewing Version
6
of
'presentation of inverse monoids and inverse semigroups'
|
[ view 'presentation of inverse monoids and inverse semigroups'
|
back to history
]
| Title of object: |
presentation of inverse monoids and inverse semigroups |
| Canonical Name: |
PresentationOfInverseMonoidsAndInverseSemigroups |
| Type: |
Definition |
| Created on: |
2006-08-21 05:59:47 |
| Modified on: |
2006-08-24 11:42:48 |
| Classification: |
msc:20M18, msc:20M05 |
| Keywords: |
Inverse Semigroups, Word Problem, Isomorphism Problem |
| Synonyms: |
presentation of inverse monoids and inverse semigroups=presentation
presentation of inverse monoids and inverse semigroups=generators and relators |
Preamble:
% 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
|
Content:
\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\cup #1^{-1}}}
\newcommand{\doubles}[1]{\cbra{#1\cup #1^{-1}}^\ast}
\newcommand{\doublep}[1]{\cbra{#1\cup #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} |
|
|
|
|
|
