Mathematical Structures: Commutative monoids

Commutative monoids

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http://mathcs.chapman.edu/structuresold/files/Commutative_monoids.pdf
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\begin{document}
\textbf{\Large Commutative monoids}

\abbreviation{CMon}
\begin{definition}
A \emph{commutative monoid} is a \href{Monoids.pdf}{monoids} $\mathbf{M}=\left\langle M,\cdot ,e\right\rangle$ such that

$\cdot$ is commutative:  $x\cdot y=y\cdot x$
\end{definition}
\begin{definition}
A \emph{commutative monoid} is a structure $\mathbf{M}=\left\langle M,\cdot ,e\right\rangle$, where $\cdot$ is an infix binary operation, called the
\emph{monoid product}, and $e$ is a constant (nullary operation), called the
\emph{identity element}, such that

$\cdot$ is commutative:  $x\cdot y=y\cdot x$

$\cdot$ is associative:  $(x\cdot y)\cdot z=x\cdot (y\cdot z)$

$e$ is an identity for $\cdot$:  $e\cdot x=x$
\end{definition}
\begin{morphisms}
Let $\mathbf{M}$ and $\mathbf{N}$ be 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}
\end{basic_results}
\begin{examples}
\begin{example}
$\langle\mathbb{N},+,0\rangle$, the natural numbers, with addition and
zero. The finitely generated free commutative monoids are direct products of this one.
\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 & decidable\\\hline
Quasiequational theory & decidable\\\hline
First-order theory & undecidable\\\hline
Locally finite & no\\\hline
Residual size & unbounded\\\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 & \\\hline
Definable principal congruences & \\\hline
Equationally def. pr. cong. & no\\\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 $n$.

$\begin{array}{lr} f(1)= &1\\ f(2)= &2\\ f(3)= &5\\ f(4)= &19\\ f(5)= &78\\ f(6)= &421\\ f(7)= &2637\\ \end{array}$
\end{finite_members}
\hyperbaseurl{http://math.chapman.edu/structures/files/}
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\begin{subclasses}\

\href{Abelian_groups.pdf}{Abelian groups}

\href{Semilattices_with_identity.pdf}{Semilattices with identity}

\end{subclasses}
\begin{superclasses}\

\href{Commutative_semigroups.pdf}{Commutative semigroups}

\href{Monoids.pdf}{Monoids}

\end{superclasses}

\begin{thebibliography}{10}

\bibitem{Ln19xx}

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\end{document}
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