Mathematical Structures: Commutative semigroups

# Commutative semigroups

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

\abbreviation{CSgrp}
\begin{definition}
A \emph{commutative semigroup} is a \href{Semigroups.pdf}{semigroups} $\mathbf{S}=\left\langle S,\cdot \right\rangle$ such that

$\cdot$ is commutative:  $xy=yx$
\end{definition}
\begin{definition}
A \emph{commutative semigroup} is a structure $\mathbf{S}=\left\langle S,\cdot \right\rangle$, where $\cdot$ is an infix binary operation, called
the \emph{semigroup product}, such that

$\cdot$ is associative:  $(xy)z=x(yz)$

$\cdot$ is commutative:  $xy=yx$
\end{definition}
\begin{morphisms}
Let $\mathbf{S}$ and $\mathbf{T}$ be commutative semigroups. A morphism from
$\mathbf{S}$ to $\mathbf{T}$ is a function $h:S\rightarrow T$ that is a
homomorphism:

$h(xy)=h(x)h(y)$

\end{morphisms}
\begin{basic_results}
\end{basic_results}
\begin{examples}
\begin{example}
$\left\langle \mathbb{N},+\right\rangle$, the natural numbers, with additition.

\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 in polynomial time\\\hline
Quasiequational theory & decidable\\\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 & \\\hline
Definable principal congruences & \\\hline
Equationally def. pr. cong. & no\\\hline
Amalgamation property & no\\\hline
Strong amalgamation property & no\\\hline
Epimorphisms are surjective & no\\\hline
\end{tabular}
\end{properties}
\end{table}
\begin{finite_members} $f(n)=$ number of members of size $n$.

$\begin{array}{lr} \href{Search_for_finite_commutative_semigroups.pdf}{Search for finite commutative semigroups} f(1)= &1\\ f(2)= &3\\ f(3)= &12\\ f(4)= &58\\ f(5)= &325\\ f(6)= &2143\\ f(7)= &17291\\ \end{array}$
\end{finite_members}
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\begin{subclasses}\

\href{Semilattices.pdf}{Semilattices}

\href{Commutative_monoids.pdf}{Commutative monoids}

\end{subclasses}
\begin{superclasses}\

\href{Semigroups.pdf}{Semigroups}

\href{Partial_commutative_semigroups.pdf}{Partial commutative semigroups}

\end{superclasses}

\begin{thebibliography}{10}

\bibitem{Ln19xx}

\end{thebibliography}

\end{document}
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