Mathematical Structures: Commutative residuated lattices

# Commutative residuated lattices

Difference (from prior minor revision) (author diff)

Changed: 32c32
 A commutative residuated lattice is a \href{Residuated_lattices.pdf}{Residuated lattices} $\mathbf{L}=\left\langle L,\vee ,\wedge ,\cdot ,e,\backslash ,/\right\rangle$ such that
 A commutative residuated lattice is a \href{Residuated_lattices.pdf}{residuated lattice} $\mathbf{L}=\langle L, \vee, \wedge, \cdot, e, \backslash, /\rangle$ such that

Changed: 35c35
 $\cdot$ is commutative: $xy=yx$
 $\cdot$ is commutative: $xy=yx$

Changed: 46,49c46,47
 $h(x\vee y)=h(x)\vee h(y)$, $h(x\wedge y)=h(x)\wedge h(y)\ \mbox{and} h(x\cdot y)=h(x)\cdot h(y)$, $h(x\backslash y)=h(x)\backslash h(y)$, $h(x/y)=h(x)/h(y)\$, $h(e)=e$
 $h(x\vee y)=h(x)\vee h(y)$, $h(x\wedge y)=h(x)\wedge h(y)$, $h(x\cdot y)=h(x)\cdot h(y)$, $h(x\backslash y)=h(x)\backslash h(y)$, $h(x/y)=h(x)/h(y)$, and $h(e)=e$

http://mathcs.chapman.edu/structuresold/files/Commutative_residuated_lattices.pdf
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\newtheorem*{morphisms}{Morphisms}
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\begin{document}
\textbf{\Large Commutative residuated lattices}

\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}

\begin{thebibliography}{10}

\bibitem{Ln19xx}

\end{thebibliography}

\end{document}
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