Commutative lattice-ordered rings

http://mathcs.chapman.edu/structuresold/files/Commutative_lattice-ordered_rings.pdf

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\begin{document}
\textbf{\Large Commutative lattice-ordered rings}
\quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Commutative_lattice-ordered_rings}{edit}

\abbreviation{CLRng}

\begin{definition}
A \emph{commutative lattice-ordered ring} is a \href{Lattice-ordered_rings.pdf}{lattice-ordered ring} $\mathbf{A}=\langle A,\vee,\wedge,+,-,0,\cdot\rangle$ 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 lattice-ordered rings. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism: 
$h(x \vee y)=h(x) \vee h(y)$,
$h(x \wedge y)=h(x) \wedge h(y)$,
$h(x + y)=h(x) + h(y)$,
$h(x \cdot y)=h(x) \cdot h(y)$.
\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                       & (value, see description) \cite{Ln19xx} \\\hline
  Equational theory               & \\\hline
  Quasiequational theory          & \\\hline
  First-order theory              & \\\hline
  Locally finite                  & \\\hline
  Residual size                   & \\\hline
  Congruence distributive         & yes \\\hline
  Congruence modular              & yes \\\hline
  Congruence $n$-permutable       & yes, $n=2$\\\hline
  Congruence regular              & yes \\\hline
  Congruence uniform              & yes \\\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} All nontrivial members are infinite.
\end{finite_members}

\begin{subclasses}\ 

  \href{Commutative f-rings.pdf}{Commutative f-rings} subvariety

\end{subclasses}

\begin{superclasses}\ 

  \href{Lattice-ordered_rings.pdf}{Lattice-ordered rings} supervariety

  \href{Abelian_lattice-ordered_groups.pdf}{Abelian lattice-ordered groups} subreduct

  \href{Commutative_rings.pdf}{Commutative rings} subreduct

\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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