Mathematical Structures: Cancellative commutative semigroups

# Cancellative commutative semigroups

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

\abbreviation{CanCSgrp}
\begin{definition}
A \emph{cancellative commutative semigroup} is a \href{Commutative_semigroups.pdf}{commutative semigroup} $\mathbf{S}=\langle S,\cdot \rangle$ such that

$\cdot$ is \emph{cancellative}:  $x\cdot z=y\cdot z\implies x=y$
\end{definition}
\begin{morphisms}
Let $\mathbf{S}$ and $\mathbf{T}$ be cancellative 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}
$\langle \mathbb{N},+\rangle$, 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 & Quasivariety\\\hline
Equational theory & \\\hline
Quasiequational theory & \\\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 $n$.

$\begin{array}{lr} f(1)= &1\\ f(2)= &\\ f(3)= &\\ f(4)= &\\ f(5)= &\\ f(6)= &\\ f(7)= &\\ \end{array}$
\end{finite_members}
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\begin{subclasses}\

\href{Cancellative_commutative_monoids.pdf}{Cancellative commutative monoids}

\end{subclasses}
\begin{superclasses}\

\href{Cancellative_semigroups.pdf}{Cancellative semigroups}

\href{Commutative_semigroups.pdf}{Commutative semigroups}

\end{superclasses}

\begin{thebibliography}{10}

\bibitem{Ln19xx}

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\end{document}
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