Almost distributive lattices

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

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

\abbreviation{ADLat}

\begin{definition}
An \emph{almost distributive lattice} is a \href{Neardistributive_lattices.pdf}{neardistributive lattice} $\mathbf{L}=\langle L,\vee,\wedge\rangle$ such that

AD$_{\wedge}$:  $v\wedge[u\vee (x\wedge[y\vee (x\wedge z)])]\le u\vee [(x\wedge[y\vee (x\wedge z)])\wedge(v\vee (x\wedge y)\vee (x\wedge z))]$

AD$_{\vee}$:  $v\vee[u\wedge (x\vee[y\wedge (x\vee z)])]\ge u\wedge [(x\vee[y\wedge (x\vee z)])\vee(v\wedge (x\vee y)\wedge (x\vee z))]$
\end{definition}

\begin{morphisms}
Let $\mathbf{L}$ and $\mathbf{M}$ be almost 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}
$D[d]=\langle D\cup\{d'\},\vee ,\wedge\rangle$, where $D$ is any distributive lattice and $d$ is an element in it that
is split into two elements $d,d'$ using Alan Day's doubling construction.
\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 & \\\hline
Quasiequational theory & \\\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 & no\\\hline
Congruence regular & no\\\hline
Congruence uniform & no\\\hline
Congruence extension property & \\\hline
Definable principal congruences & \\\hline
Equationally def. pr. cong. & \\\hline
Amalgamation property & no\\\hline
Strong amalgamation property & no\\\hline
Epimorphisms are surjective & \\\hline
\end{tabular}
\end{properties}
\end{table}

\begin{finite_members} $f(n)=$ number of members of size $n$.

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

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

\href{Distributive_lattices.pdf}{Distributive lattices} 

\end{subclasses}

\begin{superclasses}\ 

\href{Neardistributive_lattices.pdf}{Neardistributive lattices} 

\end{superclasses}

\begin{thebibliography}{10}

\bibitem{Ln19xx}

\end{thebibliography}

\end{document}
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