Commutative residuated lattices

http://mathcs.chapman.edu/structuresold/files/Commutative_residuated_lattices.pdf

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\begin{document}
\textbf{\Large Commutative residuated lattices}
\quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Commutative_residuated_lattices}{edit}

\abbreviation{CRL}
\begin{definition}
A \emph{commutative residuated lattice} is a \href{Residuated_lattices.pdf}{residuated lattice} $\mathbf{L}=\langle L, \vee, \wedge, \cdot, e, \backslash, /\rangle $ such that


$\cdot$ is commutative:  $xy=yx$


Remark: 

\end{definition}
\begin{morphisms}
Let $\mathbf{L}$ and $\mathbf{M}$ be commutative residuated lattices. A
morphism from $\mathbf{L}$ to $\mathbf{M}$ is a function $h:L\rightarrow M$
that is a homomorphism: 

$h(x\vee y)=h(x)\vee h(y)$, $h(x\wedge y)=h(x)\wedge h(y)$,
$h(x\cdot y)=h(x)\cdot h(y)$, $h(x\backslash y)=h(x)\backslash h(y)$, $h(x/y)=h(x)/h(y)$, and $h(e)=e$

\end{morphisms}
\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})

\begin{tabular}{|ll|}\hline
Classtype & Variety\\\hline
Equational theory & Decidable\\\hline
Quasiequational theory & Undecidable\\\hline
First-order theory & Undecidable\\\hline
Locally finite & No\\\hline
Residual size & Unbounded\\\hline
Congruence distributive & Yes\\\hline
Congruence modular & Yes\\\hline
Congruence n-permutable & Yes, n=2\\\hline
Congruence regular & No\\\hline
Congruence e-regular & Yes\\\hline
Congruence uniform & No\\\hline
Congruence extension property & Yes\\\hline
Definable principal congruences & No\\\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 $n$.

$\begin{array}{lr}
f(1)= &1\\
f(2)= &1\\
f(3)= &3\\
f(4)= &16\\
f(5)= &100\\
f(6)= &794\\
\end{array}$
\end{finite_members}
\hyperbaseurl{http://math.chapman.edu/structures/files/}
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\begin{subclasses}\ 

\href{Commutative_distributive_residuated_lattices.pdf}{Commutative distributive residuated lattices} 

\href{FLe-algebras.pdf}{FLe-algebras} 

\end{subclasses}
\begin{superclasses}\ 

\href{Commutative_multiplicative_lattices.pdf}{Commutative multiplicative lattices} 

\href{Commutative_residuated_join-semilattices.pdf}{Commutative residuated join-semilattices} 

\href{Commutative_residuated_meet-semilattices.pdf}{Commutative residuated meet-semilattices} 

\href{Residuated_lattices.pdf}{Residuated lattices} 

\end{superclasses}

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\bibitem{Ln19xx}

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\end{document}
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