Mathematical Structures: Commutative residuated partially ordered semigroups

Commutative residuated partially ordered semigroups

Difference (from prior major revision) (no other diffs)

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 \Large Commutative residuated partially ordered semigroups \quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Commutative_residuated_partially_ordered_semigroups}{edit}

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 \abbreviation{Abbr}
 \abbreviation{CRPoSgrp}

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 A ... is a structure $\mathbf{A}=\langle A,...\rangle$ of type $\langle ...\rangle$ such that $\langle A,...\rangle$ is a \href{Name_of_class.pdf}{name of class}
 A commutative residuated partially ordered semigroup is a \href{Residuated_partially_ordered_semigroups.pdf}{residuated partially ordered semigroup} $\mathbf{A}=\langle A, \cdot, \to, \le\rangle$ such that

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 $op_1$ is (name of property): $axiom_1$ $op_2$ is ...: $...$
 $\cdot$ is commutative: $xy=yx$

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 Let $\mathbf{A}$ and $\mathbf{B}$ be ... . A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism: $h(x ... y)=h(x) ... h(y)$
 Let $\mathbf{A}$ and $\mathbf{B}$ be commutative residuated partially ordered monoids. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a orderpreserving homomorphism: $h(x \cdot y)=h(x) \cdot h(y)$, $h(x \to y)=h(x) \to h(y)$, and $x\le y\implies h(x)\le h(y)$.

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 Classtype & (value, see description) \cite{Ln19xx} \\\hline
 Classtype & quasivariety \\\hline

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 \href{....pdf}{...} subvariety \href{....pdf}{...} expansion
 \href{Commutative_residuated_lattice-ordered_semigroups.pdf}{Commutative residuated lattice-ordered semigroups} expanded type

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 \href{....pdf}{...} supervariety
 \href{Residuated_partially_ordered_semigroups.pdf}{Residuated partially ordered semigroups} same type

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 \href{....pdf}{...} subreduct
 \href{Commutative_partially_ordered_semigroups.pdf}{Commutative partially ordered semigroups} reduced type

http://mathcs.chapman.edu/structuresold/files/Commutative_residuated_partially_ordered_semigroups.pdf
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\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}
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\begin{document}
\textbf{\Large Commutative residuated partially ordered semigroups}

\abbreviation{CRPoSgrp}

\begin{definition}
A \emph{commutative residuated partially ordered semigroup} is a \href{Residuated_partially_ordered_semigroups.pdf}{residuated partially ordered semigroup} $\mathbf{A}=\langle A, \cdot, \to, \le\rangle$ such that

$\cdot$ is \emph{commutative}:  $xy=yx$

Remark: This is a template.
If you know something about this class, click on the Edit text of this page'' link at the bottom and fill out this page.

It is not unusual to give several (equivalent) definitions. Ideally, one of the definitions would give an irredundant axiomatization that does not refer to other classes.
\end{definition}

\begin{morphisms}
Let $\mathbf{A}$ and $\mathbf{B}$ be commutative residuated partially ordered monoids. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a orderpreserving homomorphism:
$h(x \cdot y)=h(x) \cdot h(y)$,
$h(x \to y)=h(x) \to h(y)$, and $x\le y\implies h(x)\le h(y)$.
\end{morphisms}

\begin{definition}
A \emph{...} is a structure $\mathbf{A}=\langle A,...\rangle$ of type $\langle ...\rangle$ such that

$...$ is ...:  $axiom$

$...$ is ...:  $axiom$
\end{definition}

\begin{basic_results}
\end{basic_results}

\begin{examples}
\begin{example}
\end{example}
\end{examples}

\begin{table}[h]
\begin{properties} (\href{http://math.chapman.edu/cgi-bin/structures?Properties}{description})

Feel free to add or delete properties from this list. The list below may contain properties that are not relevant to the class that is being described.

\begin{tabular}{|ll|}\hline
Classtype                       & quasivariety \\\hline
Equational theory               & \\\hline
Quasiequational theory          & \\\hline
First-order theory              & \\\hline
Locally finite                  & \\\hline
Residual size                   & \\\hline
Congruence distributive         & \\\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 $n$.

$\begin{array}{lr} f(1)= &1\\ f(2)= &\\ f(3)= &\\ f(4)= &\\ f(5)= &\\ \end{array}$\qquad
$\begin{array}{lr} f(6)= &\\ f(7)= &\\ f(8)= &\\ f(9)= &\\ f(10)= &\\ \end{array}$

\end{finite_members}

\begin{subclasses}\

\href{Commutative_residuated_lattice-ordered_semigroups.pdf}{Commutative residuated lattice-ordered semigroups} expanded type

\end{subclasses}

\begin{superclasses}\

\href{Residuated_partially_ordered_semigroups.pdf}{Residuated partially ordered semigroups} same type

\href{Commutative_partially_ordered_semigroups.pdf}{Commutative partially ordered semigroups} reduced type

\end{superclasses}

\begin{thebibliography}{10}

\bibitem{Lastname19xx}
F. Lastname, \emph{Title}, Journal, \textbf{1}, 23--45 \href{http://www.ams.org/mathscinet-getitem?mr=12a:08034}{MRreview}

\end{thebibliography}

\end{document}
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