Mathematical Structures: Pointed residuated lattices

Pointed residuated lattices

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

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

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

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

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

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

Remark:

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

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

\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
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}
[http://www.chapman.edu/~jipsen/reslat/ 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 $n$.

$\begin{array}{lr} [http://www.chapman.edu/~jipsen/gap/rl.html Small residuated lattices] f(1)= &1\\ f(2)= &2\\ f(3)= &9\\ f(4)= &\\ f(5)= &\\ f(6)= &\\ f(7)= &\\ \end{array}$
\end{finite_members}
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\begin{subclasses}\

\href{Residuated_lattices.pdf}{Residuated lattices} subvariety

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

\end{subclasses}
\begin{superclasses}\

\href{Residuated_lattices.pdf}{Residuated lattices} reducts

\end{superclasses}

\begin{thebibliography}{10}

\bibitem{Ln19xx}

\end{thebibliography}

\end{document}
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