History of Cancellative commutative monoids

 
 \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 commutative monoids}
 \quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Cancellative_commutative_monoids}{edit}
 
 \abbreviation{CanCMon}
 \begin{definition}
 A \emph{cancellative commutative monoid} is a \href{Cancellative_monoids.pdf}{cancellative monoids} $\mathbf{M}=\left\langle M,\cdot
 ,e\right\rangle $ such that
 
 $\cdot $ is commutative:  $x\cdot y=y\cdot x$
 \end{definition}
 \begin{morphisms}
 Let $\mathbf{M}$ and $\mathbf{N}$ be cancellative commutative 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 commutative free monoids are cancellative.
 
 All finite commutative (left or right) cancellative monoids are reducts of \href{Abelian_groups.pdf}{abelian 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 $n$.
 
 $\begin{array}{lr}
 f(1)= &1\\
 f(2)= &1\\
 f(3)= &1\\
 f(4)= &2\\
 f(5)= &1\\
 f(6)= &1\\
 f(7)= &1\\
 \end{array}$
 \end{finite_members}
 \hyperbaseurl{http://math.chapman.edu/structures/files/}
 \parskip0pt
 \begin{subclasses}\ 
 
 \href{Abelian_groups.pdf}{Abelian groups} 
 
 \href{Cancellative_commutative_residuated_lattices.pdf}{Cancellative commutative residuated lattices} 
 
 \end{subclasses}
 \begin{superclasses}\ 
 
 \href{Cancellative_commutative_semigroups.pdf}{Cancellative commutative semigroups} 
 
 \href{Cancellative_monoids.pdf}{Cancellative monoids} 
 
 \href{Commutative_monoids.pdf}{Commutative monoids} 
 
 \end{superclasses}
 
 \begin{thebibliography}{10}
 
 \bibitem{Ln19xx}
 
 \end{thebibliography}
 
 \end{document}
 %