Residuated lattices

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

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\newtheorem*{morphisms}{Morphisms}
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\begin{document}
\textbf{\Large Residuated lattices}
\quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Residuated_lattices}{edit}

\abbreviation{RL}

\begin{definition}
A \emph{residuated lattice} is a structure $\mathbf{L}=\langle L, \vee, \wedge, \cdot, e, \backslash, /\rangle$ of type $\langle
2,2,2,0,2,2\rangle$ such that

$\langle L, \cdot, e\rangle$ is a \href{Monoids.pdf}{monoid}

$\langle L, \vee, \wedge\rangle$ is a \href{Lattices.pdf}{lattice}

$\backslash$ is the left residual of $\cdot$: $y\leq x\backslash z\Longleftrightarrow xy\leq z$

$/$ is the right residual of $\cdot$: $x\leq z/y\Longleftrightarrow xy\leq z$
\end{definition}

\begin{morphisms}
Let $\mathbf{L}$ and $\mathbf{M}$ be 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               & decidable \cite{OK1985} \href{http://www.chapman.edu/~jipsen/reslat/implementation}{implementation}\\\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 $n$.

$\begin{array}{lr}
f(1)= &1\\
f(2)= &1\\
f(3)= &3\\
f(4)= &20\\
f(5)= &149\\
f(6)= &1488\\
f(7)= &18554\\
\end{array}$

\href{http://www.chapman.edu/~jipsen/gap/rl.html}{Small residuated lattices}
\end{finite_members}

\hyperbaseurl{http://math.chapman.edu/structures/files/}
\begin{subclasses}\ 

\href{Commutative_residuated_lattices.pdf}{Commutative residuated lattices} 

\href{Distributive_residuated_lattices.pdf}{Distributive residuated lattices} 

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

\href{Integral_residuated_lattices.pdf}{Integral residuated lattices} 

\end{subclasses}
\begin{superclasses}\ 

\href{Multiplicative_lattices.pdf}{Multiplicative lattices} 

\href{Residuated_join-semilattices.pdf}{Residuated join-semilattices} 

\href{Residuated_meet-semilattices.pdf}{Residuated meet-semilattices} 

\end{superclasses}

\begin{thebibliography}{10}

\bibitem{OK1985}
Hiroakira Ono, Yuichi Komori, \emph{Logics without the contraction rule},
J. Symbolic Logic, \textbf{50}, 1985, 169--201 \href{http://www.ams.org/mathscinet-getitem?mr=87a:03053}{MRreview}\href{http://www.emis.de/MATH-item?0583.03018}{ZMATH}

\end{thebibliography}

\end{document}
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