Mathematical Structures: History of Cancellative monoids

# History of Cancellative monoids

 %%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 Cancellative monoids} \quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Cancellative_monoids}{edit} \abbreviation{CanMon} \begin{definition} A \emph{cancellative monoid} is a \href{Monoids.pdf}{monoids} $\mathbf{M}=\left\langle M,\cdot ,e\right\rangle$ such that $\cdot$ is left cancellative: $z\cdot x=z\cdot y\implies x=y$ $\cdot$ is right cancellative: $x\cdot z=y\cdot z\implies x=y$ \end{definition} \begin{morphisms} Let $\mathbf{M}$ and $\mathbf{N}$ be cancellative monoids. A morphism from $\mathbf{M}$ to $\mathbf{N}$ is a function $h:M\rightarrow N$ that is a homomorphism: $h(x\cdot y)=h(x)\cdot h(y)$, $h(e)=e$ \end{morphisms} \begin{basic_results} All free monoids are cancellative. All finite (left or right) cancellative monoids are reducts of \href{Groups.pdf}{groups}. \end{basic_results} \begin{examples} \begin{example} $\langle\mathbb{N},+,0\rangle$, the natural numbers, with addition and zero. \end{example} \end{examples} \begin{table}[h] \begin{properties} (\href{http://math.chapman.edu/cgi-bin/structures?Properties}{description}) \begin{tabular}{|ll|}\hline Classtype & quasivariety\\\hline Equational theory & \\\hline Quasiequational theory & \\\hline First-order theory & undecidable\\\hline Locally finite & no\\\hline Residual size & unbounded\\\hline Congruence distributive & no\\\hline Congruence modular & \\\hline Congruence n-permutable & \\\hline Congruence regular & \\\hline Congruence uniform & \\\hline Congruence extension property & \\\hline Definable principal congruences & \\\hline Equationally def. pr. cong. & \\\hline Amalgamation property & \\\hline Strong amalgamation property & \\\hline Epimorphisms are surjective & \\\hline \end{tabular} \end{properties} \end{table} \begin{finite_members} $f(n)=$ number of members of size $n$. $\begin{array}{lr} f(1)= &1\\ f(2)= &1\\ f(3)= &1\\ f(4)= &2\\ f(5)= &1\\ f(6)= &2\\ f(7)= &1\\ \end{array}$ \end{finite_members} \hyperbaseurl{http://math.chapman.edu/structures/files/} \parskip0pt \begin{subclasses}\ \href{Groups.pdf}{Groups} \href{Cancellative_residuated_lattices.pdf}{Cancellative residuated lattices} \end{subclasses} \begin{superclasses}\ \href{Cancellative_semigroups.pdf}{Cancellative semigroups} \href{Monoids.pdf}{Monoids} \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 Cancellative monoids} \quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Cancellative_monoids}{edit} \abbreviation{CanMon} \begin{definition} A \emph{cancellative monoid} is a \href{Monoids.pdf}{monoid} $\mathbf{M}=\langle M, \cdot, e\rangle$ such that $\cdot$ is left cancellative: $z\cdot x=z\cdot y\implies x=y$ $\cdot$ is right cancellative: $x\cdot z=y\cdot z\implies x=y$ \end{definition} \begin{morphisms} Let $\mathbf{M}$ and $\mathbf{N}$ be cancellative monoids. A morphism from $\mathbf{M}$ to $\mathbf{N}$ is a function $h:M\rightarrow N$ that is a homomorphism: $h(x\cdot y)=h(x)\cdot h(y)$, $h(e)=e$ \end{morphisms} \begin{basic_results} All free monoids are cancellative. All finite (left or right) cancellative monoids are reducts of \href{Groups.pdf}{groups}. \end{basic_results} \begin{examples} \begin{example} $\langle\mathbb{N},+,0\rangle$, the natural numbers, with addition and zero. \end{example} \end{examples} \begin{table}[h] \begin{properties} (\href{http://math.chapman.edu/cgi-bin/structures?Properties}{description}) \begin{tabular}{|ll|}\hline Classtype & quasivariety\\\hline Equational theory & \\\hline Quasiequational theory & \\\hline First-order theory & undecidable\\\hline Locally finite & no\\\hline Residual size & unbounded\\\hline Congruence distributive & no\\\hline Congruence modular & \\\hline Congruence n-permutable & \\\hline Congruence regular & \\\hline Congruence uniform & \\\hline Congruence extension property & \\\hline Definable principal congruences & \\\hline Equationally def. pr. cong. & \\\hline Amalgamation property & \\\hline Strong amalgamation property & \\\hline Epimorphisms are surjective & \\\hline \end{tabular} \end{properties} \end{table} \begin{finite_members} $f(n)=$ number of members of size $n$. $\begin{array}{lr} f(1)= &1\\ f(2)= &1\\ f(3)= &1\\ f(4)= &2\\ f(5)= &1\\ f(6)= &2\\ f(7)= &1\\ \end{array}$ \end{finite_members} \hyperbaseurl{http://math.chapman.edu/structures/files/} \begin{subclasses}\ \href{Groups.pdf}{Groups} \href{Cancellative_residuated_lattices.pdf}{Cancellative residuated lattices} \end{subclasses} \begin{superclasses}\ \href{Cancellative_semigroups.pdf}{Cancellative semigroups} \href{Monoids.pdf}{Monoids} \end{superclasses} \begin{thebibliography}{10} \bibitem{Ln19xx} \end{thebibliography} \end{document} %