Mathematical Structures: Distributive lattices

# Distributive lattices

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

\abbreviation{DLat}
\begin{definition}
A \emph{distributive lattice} is a lattice $\mathbf{L}=\langle L,\vee ,\wedge\rangle$ such that

$\wedge$ distributes over $\vee$:  $x\wedge (y\vee z) = (x\wedge y) \vee (x\wedge z)$
\end{definition}

\begin{definition}
A \emph{distributive lattice} is a lattice $\mathbf{L}=\langle L,\vee ,\wedge\rangle$ such that

$\vee$ distributes over $\wedge$:  $x\vee (y\wedge z) = (x\vee y) \wedge (x\vee z)$
\end{definition}

\begin{definition}
A \emph{distributive lattice} is a lattice $\mathbf{L}=\langle L,\vee ,\wedge \rangle$ such that

$(x\wedge y) \vee (x\wedge z) \vee (y\wedge z) = (x\vee y) \wedge (x\vee z) \wedge (y\vee z)$
\end{definition}

\begin{definition}
A \emph{distributive lattice} is a lattice $\mathbf{L}=\left\langle L,\vee ,\wedge \right\rangle$ such that $\mathbf{L}$ has no sublattice isomorphic
to the diamond $\mathbf{M}_{3}$ or the pentagon $\mathbf{N}_{5}$
\end{definition}

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

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

\begin{basic_results}
\end{basic_results}

\begin{examples}
\begin{example}
$\left\langle P(S),\cup ,\cap ,\subseteq \right\rangle$, the collection of
subsets of a sets $S$, ordered by inclusion.
\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. & yes, $\begin{array}{c}\langle c,d\rangle\in \text{Cg}(a,b)\iff \\ (a\wedge b)\wedge c=(a\wedge b)\wedge d\\ (a\vee b)\vee c=(a\vee b)\vee d\end{array}$\\\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 $n$.

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

Values known up to size 49 \cite{ErneHeitigReinhold2002}
\end{finite_members}
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\begin{subclasses}\

\href{One-element_algebras.pdf}{One-element algebras}

\href{Bounded_distributive_lattices.pdf}{Bounded distributive lattices}

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

\end{subclasses}
\begin{superclasses}\

\href{Modular_lattices.pdf}{Modular lattices}

\href{Semidistributive_lattices.pdf}{Semidistributive lattices}

\end{superclasses}

\begin{thebibliography}{10}

\bibitem{ErneHeitigReinhold2002}
M. Ern\'e, J. Heitzig, J. Reinhold,
\emph{On the number of distributive lattices},
Electronic J. Combinatorics 9 (2002), no. 1, Research Paper 24, 23 pp.

\end{thebibliography}

\end{document}
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