Bounded residuated lattices

http://math.chapman.edu/structuresold/files/Bounded_residuated_lattices.pdf

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Transcript written on Bounded_residuated_lattices.log.

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

\abbreviation{RLat$_b$}

\begin{definition}
A \emph{bounded residuated lattice} is a \href{Residuated_lattices.pdf}{residuated lattice}
that is bounded:

$\bot$ is the least element:  $\bot\vee x=x$

$\top$ is the greatest element:  $\top\vee x=\top$
\end{definition}

\begin{morphisms}
Let $\mathbf{A}$ and $\mathbf{B}$ be bounded residuated lattices. 
A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a residuated lattice homomorphism $h:A\rightarrow B$ that preserves the bounds: 
$h(\bot)=\bot$ and $h(\top)=\top$.
\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              & 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 $n$.

$\begin{array}{lr}
  f(1)= &1\\
  f(2)= &\\
  f(3)= &\\
  f(4)= &\\
  f(5)= &\\
\end{array}$\qquad
$\begin{array}{lr}
  f(6)= &\\
  f(7)= &\\
  f(8)= &\\
  f(9)= &\\
  f(10)= &\\
\end{array}$

\end{finite_members}

\begin{subclasses}\ 

  \href{....pdf}{...} subvariety

  \href{....pdf}{...} expansion

\end{subclasses}

\begin{superclasses}\ 

  \href{....pdf}{...} supervariety

  \href{....pdf}{...} subreduct

\end{superclasses}

\begin{thebibliography}{10}

\bibitem{Ln19xx}
F. Lastname, \emph{Title}, Journal, \textbf{1}, 23--45 \href{http://www.ams.org/mathscinet-getitem?mr=12a:08034}{MRreview} 

\end{thebibliography}

\end{document}
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