Mathematical Structures: Distributive residuated lattices

# Distributive residuated lattices

http://mathcs.chapman.edu/structuresold/files/Distributive_residuated_lattices.pdf
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\begin{document}
\textbf{\Large Distributive residuated lattices}

\abbreviation{DRL}
\begin{definition}
A \emph{distributive residuated lattice} is a residuated lattice $\mathbf{L}=\langle L, \vee, \wedge, \cdot, e, \backslash, /\rangle$ such that

$\vee, \wedge$ are distributive:  $x\wedge(y\vee z) =(x\wedge y) \vee (x\wedge z)$

Remark:

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

$h(x\vee y)=h(x)\vee h(y)$, $h(x\wedge y)=h(x)\wedge h(y)$, $h(x\cdot y)=h(x)\cdot h(y)$, $h(x\backslash y)=h(x)\backslash h(y)$, $h(x/y)=h(x)/h(y)$, $h(e)=e$
\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 & \\\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 & no\\\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 $n$.

$\begin{array}{lr} f(1)= &1\\ f(2)= &1\\ f(3)= &3\\ f(4)= &16\\ f(5)= &\\ f(6)= &\\ \end{array}$
\end{finite_members}
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\begin{subclasses}\

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

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

\end{subclasses}
\begin{superclasses}\

\href{Distributive_multiplicative_lattices.pdf}{Distributive multiplicative lattices}

\href{Residuated_lattices.pdf}{Residuated lattices}

\end{superclasses}

\begin{thebibliography}{10}

\bibitem{Ln19xx}

\end{thebibliography}

\end{document}
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