Mathematical Structures: Semidistributive lattices

# Semidistributive lattices

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

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

SD$_{\wedge}$:  $x\wedge y=x\wedge z\implies x\wedge y=x\wedge(y\vee z)$

SD$_{\vee}$:  $x\vee y=x\vee z\implies x\vee y=x\vee(y\wedge z)$
\end{definition}

\begin{morphisms}
Let $\mathbf{L}$ and $\mathbf{M}$ be semidistributive 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 & quasivariety\\\hline
Equational theory & \\\hline
Quasiequational theory & \\\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 & \\\hline
Definable principal congruences & \\\hline
Equationally def. pr. cong. & \\\hline
Amalgamation property & no\\\hline
Strong amalgamation property & no\\\hline
Epimorphisms are surjective & \\\hline
Locally finite & no\\\hline
Residual size & unbounded\\\hline
\end{tabular}
\end{properties}
\end{table}
\begin{finite_members} $f(n)=$ number of members of size $n$.

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

\href{Neardistributive_lattices.pdf}{Neardistributive lattices}

\end{subclasses}
\begin{superclasses}\

\href{Join-semidistributive_lattices.pdf}{Join-semidistributive lattices}

\href{Meet-semidistributive_lattices.pdf}{Meet-semidistributive lattices}

\end{superclasses}

\begin{thebibliography}{10}

\bibitem{Ln19xx}

\end{thebibliography}

\end{document}
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