Bounded distributive lattices

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

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

\abbreviation{BDLat}

\begin{definition}
A \emph{bounded distributive lattice} is a structure $\mathbf{L}=\left\langle L,\vee ,0,\wedge ,1\right\rangle $ such that

$\left\langle L,\vee ,\wedge \right\rangle $ is a 
\href{Distributive_lattices}{distributive lattice}

$0$ is the least element:  $0\leq x$

$1$ is the greatest element:  $x\leq 1$
\end{definition}

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

$h(x\vee y)=h(x)\vee h(y)$, $h(x\wedge y)=h(x)\wedge h(y)$, $h(0)=0$, $h(1)=1$
\end{morphisms}

\begin{basic_results}
\end{basic_results}

\begin{examples}
\begin{example}
$\langle \mathcal P(S), \cup, \emptyset, \cap, S\rangle$, the collection
of subsets of a set $S$, with union, empty set, intersection, and the whole
set $S$.

\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 & decidable\\\hline
First-order theory & undecidable\\\hline
Congruence distributive & yes\\\hline
Congruence modular & yes\\\hline
Congruence n-permutable & no\\\hline
Congruence regular & no\\\hline
Congruence uniform & no\\\hline
Congruence extension property & yes\\\hline
Definable principal congruences & no\\\hline
Equationally def. pr. cong. & no\\\hline
Amalgamation property & yes\\\hline
Strong amalgamation property & no\\\hline
Epimorphisms are surjective & no\\\hline
Locally finite & yes\\\hline
Residual size & 2\\\hline
\end{tabular}
\end{properties}
\end{table}
\begin{finite_members} $f(n)=$ number of members of size $n$.

$\begin{array}{lr}
f(1)= &1\\
f(2)= &1\\
f(3)= &1\\
f(4)= &2\\
f(5)= &3\\
\end{array}$\qquad
$\begin{array}{lr}
f(6)= &5\\
f(7)= &8\\
f(8)= &15\\
f(9)= &26\\
f(10)= &47\\
\end{array}$\qquad
$\begin{array}{lr}
f(11)= &82\\
f(12)= &151\\
f(13)= &269\\
f(14)= &494\\
f(15)= &891\\
\end{array}$\qquad
$\begin{array}{lr}
f(16)= &1639\\
f(17)= &2978\\
f(18)= &5483\\
f(19)= &10006\\
f(20)= &18428\\
\end{array}$

Values known up to size 49 \cite{EHR2002}.
\end{finite_members}
\hyperbaseurl{http://math.chapman.edu/structures/files/}
\parskip0pt
\begin{subclasses}\ 

\href{Boolean_algebras.pdf}{Boolean algebras} 

\href{Complete_distributive_lattices.pdf}{Complete distributive lattices} 

\end{subclasses}
\begin{superclasses}\ 

\href{Distributive_lattices.pdf}{Distributive lattices} 

\href{Bounded_modular_lattices.pdf}{Bounded modular lattices} 

\end{superclasses}

\begin{thebibliography}{10}

\bibitem{EHR2002}
Marcel Erne, Jobst Heitzig and J\"urgen Reinhold, \emph{On the number of distributive lattices}, Electron. J. Combin.,
\textbf{9}, 2002,Research Paper 24, 23 pp. (electronic)
\href{http://www.ams.org/mathscinet-getitem?mr=2003c:05012}{MRreview}

\end{thebibliography}

\end{document}
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