Mathematical Structures: Cancellative commutative monoids

# Cancellative commutative monoids

http://mathcs.chapman.edu/structuresold/files/Cancellative_commutative_monoids.pdf
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\begin{document}
\textbf{\Large Cancellative commutative monoids}

\abbreviation{CanCMon}
\begin{definition}
A \emph{cancellative commutative monoid} is a \href{Cancellative_monoids.pdf}{cancellative monoid} $\mathbf{M}=\left\langle M,\cdot ,e\right\rangle$ 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/}
\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}
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