Mathematical Structures: History of Clifford semigroups

# History of Clifford semigroups

 %%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 Clifford semigroups} \quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Clifford_semigroups}{edit} \abbreviation{CliffSgrp} \begin{definition} A \emph{Clifford semigroup} is an \href{Inverse_semigroups.pdf}{inverse semigroups} $\mathbf{S}=\langle S,\cdot,^{-1}\rangle$ that is also \href{Completely_regular_semigroups.pdf}{completely regular semigroups}. \end{definition} \begin{definition} A \emph{Clifford semigroup} is a structure $\mathbf{S}=\langle S,\cdot,^{-1}\rangle$ such that $\cdot$ is associative: $(xy)z=x(yz)$ $^{-1}$ is an inverse: $xx^{-1}x=x$, $(x^{-1})^{-1}=x$ $xx^{-1}=x^{-1}x$, $xx^{-1}y^{-1}y=y^{-1}yxx^{-1}$, $xx^{-1}=x^{-1}x$ \end{definition} \begin{morphisms} Let $\mathbf{S}$ and $\mathbf{T}$ be Clifford 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} \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 & No\\\hline Strong amalgamation property & No\\\hline Epimorphisms are surjective & Yes\\\hline \end{tabular} \end{properties} \end{table} \begin{finite_members} $f(n)=$ number of members of size $n$. $\begin{array}{lr} f(1)= &1\\ f(2)= &\\ f(3)= &\\ f(4)= &\\ f(5)= &\\ f(6)= &\\ f(7)= &\\ \end{array}$ \end{finite_members} \hyperbaseurl{http://math.chapman.edu/structures/files/} \parskip0pt \begin{subclasses}\ \href{Groups.pdf}{Groups} \end{subclasses} \begin{superclasses}\ \href{Completely_regular_semigroups.pdf}{Completely regular semigroups} \href{Inverse_semigroups.pdf}{Inverse 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 Clifford semigroups} \quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Clifford_semigroups}{edit} \abbreviation{CliffSgrp} \begin{definition} A \emph{Clifford semigroup} is an \href{Inverse_semigroups.pdf}{inverse semigroups} $\mathbf{S}=\langle S,\cdot,^{-1}\rangle$ that is also \href{Completely_regular_semigroups.pdf}{completely regular semigroups}. \end{definition} \begin{definition} A \emph{Clifford semigroup} is a structure $\mathbf{S}=\langle S,\cdot,^{-1}\rangle$ such that $\cdot$ is associative: $(xy)z=x(yz)$ $^{-1}$ is an inverse: $xx^{-1}x=x$, $(x^{-1})^{-1}=x$ $xx^{-1}=x^{-1}x$, $xx^{-1}y^{-1}y=y^{-1}yxx^{-1}$, $xx^{-1}=x^{-1}x$ \end{definition} \begin{morphisms} Let $\mathbf{S}$ and $\mathbf{T}$ be Clifford 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} \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 & No\\\hline Strong amalgamation property & No\\\hline Epimorphisms are surjective & Yes\\\hline \end{tabular} \end{properties} \end{table} \begin{finite_members} $f(n)=$ number of members of size $n$. $\begin{array}{lr} f(1)= &1\\ f(2)= &\\ f(3)= &\\ f(4)= &\\ f(5)= &\\ f(6)= &\\ f(7)= &\\ \end{array}$ \end{finite_members} \hyperbaseurl{http://math.chapman.edu/structures/files/} \parskip0pt \begin{subclasses}\ \href{Groups.pdf}{Groups} \end{subclasses} \begin{superclasses}\ \href{Completely_regular_semigroups.pdf}{Completely regular semigroups} \href{Inverse_semigroups.pdf}{Inverse semigroups} \end{superclasses} \begin{thebibliography}{10} \bibitem{Ln19xx} \end{thebibliography} \end{document} %