Mathematical Structures: Inverse semigroups

# Inverse semigroups

Difference (from prior major revision) (no other diffs)

Changed: 1,118c1,118
 %%run pdflatex % \documentclass[12pt]{amsart} \usepackage[pdfpagemode=Fullscreen,pdfstartview=FitBH]{hyperref} \parindent=0pt \parskip=5pt \addtolength{\oddsidemargin}{-.5in} \addtolength{\evensidemargin}{-.5in} \addtolength{\textwidth}{1in} \theoremstyle{definition} \newtheorem{definition}{Definition} \newtheorem*{morphisms}{Morphisms} \newtheorem*{basic_results}{Basic Results} \newtheorem*{examples}{Examples} \newtheorem{example}{} \newtheorem*{properties}{Properties} \newtheorem*{finite_members}{Finite Members} \newtheorem*{subclasses}{Subclasses} \newtheorem*{superclasses}{Superclasses} \newcommand{\abbreviation}[1]{\textbf{Abbreviation: #1}} \pagestyle{myheadings}\thispagestyle{myheadings} \markboth{\today}{math.chapman.edu/structures} \begin{document} \textbf{\Large Inverse semigroups} \quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Inverse_semigroups}{edit} \abbreviation{InvSgrp} \begin{definition} An \emph{inverse semigroup} is a structure $\mathbf{S}=\langle S,\cdot,^{-1}\rangle$ such that $\cdot$ is associative: $(xy)z=x(yz)$ $^{-1}$ is an inverse: $xx^{-1}x=x$, $(x^{-1})^{-1}=x$ idempotents commute: $xx^{-1}y^{-1}y=y^{-1}yxx^{-1}$ \end{definition} \begin{morphisms} Let $\mathbf{S}$ and $\mathbf{T}$ be inverse semigroups. A morphism from $\mathbf{S}$ to $\mathbf{T}$ is a function $h:S\rightarrow T$ that is a homomorphism: $h(xy)=h(x)h(y)$, $h(x^{-1})=h(x)^{-1}$ \end{morphisms} \begin{basic_results} \end{basic_results} \begin{examples} \begin{example} $\langle I_X,\circ,^{-1}\rangle$, the \emph{symmetric inverse semigroup} of all one-to-one partial functions on a set $X$, with composition and function inverse. Every inverse semigroup can be embedded in a symmetric inverse semigroup. \end{example} \end{examples} \begin{table}[h] \begin{properties} (\href{http://math.chapman.edu/cgi-bin/structures?Properties}{description}) \begin{tabular}{|ll|}\hline Classtype & Variety\\\hline Equational theory & \\\hline Quasiequational theory & \\\hline First-order theory & \\\hline Locally finite & No\\\hline Residual size & \\\hline Congruence distributive & No\\\hline Congruence modular & No\\\hline Congruence n-permutable & No\\\hline Congruence regular & No\\\hline Congruence uniform & No\\\hline Congruence extension property & No\\\hline Definable principal congruences & \\\hline Equationally def. pr. cong. & No\\\hline Amalgamation property & Yes\\\hline Strong amalgamation property & Yes\\\hline Epimorphisms are surjective & Yes\\\hline \end{tabular} \end{properties} \end{table} \begin{finite_members} $f(n)=$ number of members of size $n$. $\begin{array}{lr} f(1)= &1\\ f(2)= &2\\ f(3)= &5\\ f(4)= &52\\ f(5)= &208\\ f(6)= &911\\ f(7)= &\\ \end{array}$ \end{finite_members} \hyperbaseurl{http://math.chapman.edu/structures/files/} \parskip0pt \begin{subclasses}\ \href{Groups.pdf}{Groups} \href{Commutative_inverse_semigroups.pdf}{Commutative inverse semigroups} \end{subclasses} \begin{superclasses}\ \href{Semigroups.pdf}{Semigroups} \end{superclasses} \begin{thebibliography}{10} \bibitem{Ln19xx} \end{thebibliography} \end{document} %
 %%run pdflatex % \documentclass[12pt]{amsart} \usepackage[pdfpagemode=Fullscreen,pdfstartview=FitBH]{hyperref} \parindent=0pt \parskip=5pt \addtolength{\oddsidemargin}{-.5in} \addtolength{\evensidemargin}{-.5in} \addtolength{\textwidth}{1in} \theoremstyle{definition} \newtheorem{definition}{Definition} \newtheorem*{morphisms}{Morphisms} \newtheorem*{basic_results}{Basic Results} \newtheorem*{examples}{Examples} \newtheorem{example}{} \newtheorem*{properties}{Properties} \newtheorem*{finite_members}{Finite Members} \newtheorem*{subclasses}{Subclasses} \newtheorem*{superclasses}{Superclasses} \newcommand{\abbreviation}[1]{\textbf{Abbreviation: #1}} \pagestyle{myheadings}\thispagestyle{myheadings} \markboth{\today}{math.chapman.edu/structures} \begin{document} \textbf{\Large Inverse semigroups} \quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Inverse_semigroups}{edit} \abbreviation{InvSgrp} \begin{definition} An \emph{inverse semigroup} is a structure $\mathbf{S}=\langle S,\cdot,^{-1}\rangle$ such that $\cdot$ is associative: $(xy)z=x(yz)$ $^{-1}$ is an inverse: $xx^{-1}x=x$, $(x^{-1})^{-1}=x$ idempotents commute: $xx^{-1}y^{-1}y=y^{-1}yxx^{-1}$ \end{definition} \begin{morphisms} Let $\mathbf{S}$ and $\mathbf{T}$ be inverse semigroups. A morphism from $\mathbf{S}$ to $\mathbf{T}$ is a function $h:S\rightarrow T$ that is a homomorphism: $h(xy)=h(x)h(y)$, $h(x^{-1})=h(x)^{-1}$ \end{morphisms} \begin{basic_results} \end{basic_results} \begin{examples} \begin{example} $\langle I_X,\circ,^{-1}\rangle$, the \emph{symmetric inverse semigroup} of all one-to-one partial functions on a set $X$, with composition and function inverse. Every inverse semigroup can be embedded in a symmetric inverse semigroup. \end{example} \end{examples} \begin{table}[h] \begin{properties} (\href{http://math.chapman.edu/cgi-bin/structures?Properties}{description}) \begin{tabular}{|ll|}\hline Classtype & Variety\\\hline Equational theory & \\\hline Quasiequational theory & \\\hline First-order theory & \\\hline Locally finite & No\\\hline Residual size & \\\hline Congruence distributive & No\\\hline Congruence modular & No\\\hline Congruence n-permutable & No\\\hline Congruence regular & No\\\hline Congruence uniform & No\\\hline Congruence extension property & No\\\hline Definable principal congruences & \\\hline Equationally def. pr. cong. & No\\\hline Amalgamation property & Yes\\\hline Strong amalgamation property & Yes\\\hline Epimorphisms are surjective & Yes\\\hline \end{tabular} \end{properties} \end{table} \begin{finite_members} $f(n)=$ number of members of size $n$. $\begin{array}{lr} f(1)= &1\\ f(2)= &2\\ f(3)= &5\\ f(4)= &52\\ f(5)= &208\\ f(6)= &911\\ f(7)= &\\ \end{array}$ \end{finite_members} \hyperbaseurl{http://math.chapman.edu/structures/files/} \parskip0pt \begin{subclasses}\ \href{Groups.pdf}{Groups} \href{Commutative_inverse_semigroups.pdf}{Commutative inverse semigroups} \end{subclasses} \begin{superclasses}\ \href{Semigroups.pdf}{Semigroups} \end{superclasses} \begin{thebibliography}{10} \bibitem{Ln19xx} \end{thebibliography} \end{document} %

http://mathcs.chapman.edu/structuresold/files/Inverse_semigroups.pdf
%%run pdflatex

%


\documentclass[12pt]{amsart}
\usepackage[pdfpagemode=Fullscreen,pdfstartview=FitBH]{hyperref}
\parindent=0pt
\parskip=5pt
\theoremstyle{definition}
\newtheorem{definition}{Definition}
\newtheorem*{morphisms}{Morphisms}
\newtheorem*{basic_results}{Basic Results}
\newtheorem*{examples}{Examples}
\newtheorem{example}{}
\newtheorem*{properties}{Properties}
\newtheorem*{finite_members}{Finite Members}
\newtheorem*{subclasses}{Subclasses}
\newtheorem*{superclasses}{Superclasses}
\newcommand{\abbreviation}[1]{\textbf{Abbreviation: #1}}
\markboth{\today}{math.chapman.edu/structures}

\begin{document}
\textbf{\Large Inverse semigroups}

\abbreviation{InvSgrp}
\begin{definition}
An \emph{inverse semigroup} is a structure $\mathbf{S}=\langle S,\cdot,^{-1}\rangle$ such that

$\cdot$ is associative:  $(xy)z=x(yz)$

$^{-1}$ is an inverse:  $xx^{-1}x=x$, $(x^{-1})^{-1}=x$

idempotents commute:  $xx^{-1}y^{-1}y=y^{-1}yxx^{-1}$
\end{definition}
\begin{morphisms}
Let $\mathbf{S}$ and $\mathbf{T}$ be inverse semigroups. A morphism from
$\mathbf{S}$ to $\mathbf{T}$ is a function $h:S\rightarrow T$ that is a
homomorphism:

$h(xy)=h(x)h(y)$, $h(x^{-1})=h(x)^{-1}$

\end{morphisms}
\begin{basic_results}
\end{basic_results}
\begin{examples}
\begin{example}
$\langle I_X,\circ,^{-1}\rangle$, the \emph{symmetric inverse semigroup} of all one-to-one partial functions on a set $X$, with
composition and function inverse. Every inverse semigroup can be embedded in a symmetric inverse semigroup.
\end{example}
\end{examples}
\begin{table}[h]
\begin{properties} (\href{http://math.chapman.edu/cgi-bin/structures?Properties}{description})

\begin{tabular}{|ll|}\hline
Classtype & Variety\\\hline
Equational theory & \\\hline
Quasiequational theory & \\\hline
First-order theory & \\\hline
Locally finite & No\\\hline
Residual size & \\\hline
Congruence distributive & No\\\hline
Congruence modular & No\\\hline
Congruence n-permutable & No\\\hline
Congruence regular & No\\\hline
Congruence uniform & No\\\hline
Congruence extension property & No\\\hline
Definable principal congruences & \\\hline
Equationally def. pr. cong. & No\\\hline
Amalgamation property & Yes\\\hline
Strong amalgamation property & Yes\\\hline
Epimorphisms are surjective & Yes\\\hline
\end{tabular}
\end{properties}
\end{table}
\begin{finite_members} $f(n)=$ number of members of size $n$.

$\begin{array}{lr} f(1)= &1\\ f(2)= &2\\ f(3)= &5\\ f(4)= &52\\ f(5)= &208\\ f(6)= &911\\ f(7)= &\\ \end{array}$
\end{finite_members}
\hyperbaseurl{http://math.chapman.edu/structures/files/}
\parskip0pt
\begin{subclasses}\

\href{Groups.pdf}{Groups}

\href{Commutative_inverse_semigroups.pdf}{Commutative inverse semigroups}

\end{subclasses}
\begin{superclasses}\

\href{Semigroups.pdf}{Semigroups}

\end{superclasses}

\begin{thebibliography}{10}

\bibitem{Ln19xx}

\end{thebibliography}

\end{document}
%