Mathematical Structures: Boolean lattices

# Boolean lattices

Showing revision 4
http://mathcs.chapman.edu/structuresold/files/Boolean_lattices.pdf
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\begin{document}
\textbf{\Large Boolean lattices}

\abbreviation{BoolLat}
\begin{definition}
A \emph{Boolean lattice} is a
\href{Bounded_distributive_lattices.pdf}{bounded distributive lattice}
$\mathbf{L}=\left\langle L,\vee ,0,\wedge ,1\right\rangle$ such that

every element has a complement:  $\exists y(x\vee y=1\mbox{ and }x\wedge y=0)$
\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
bounded lattice 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}
$\left\langle P(S),\cup ,\varnothing ,\cap ,S\right\rangle$, 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 & first-order\\\hline
Equational theory & decidable\\\hline
Quasiequational theory & decidable\\\hline
First-order theory & decidable\\\hline
Congruence distributive & yes\\\hline
Congruence modular & yes\\\hline
Congruence n-permutable & yes\\\hline
Congruence regular & yes\\\hline
Congruence uniform & yes\\\hline
Congruence extension property & yes\\\hline
Definable principal congruences & yes\\\hline
Equationally def. pr. cong. & \\\hline
Amalgamation property & \\\hline
Strong amalgamation property & \\\hline
Epimorphisms are surjective & \\\hline
Locally finite & yes\\\hline
Residual size & \\\hline
\end{tabular}
\end{properties}
\end{table}

\begin{finite_members}
Any finite member is a power of the 2-element Boolean lattice.
\end{finite_members}

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

\href{Boolean_algebras.pdf}{Boolean algebras}

\end{subclasses}
\begin{superclasses}\

\href{Complemented_modular_lattices.pdf}{Complemented modular lattices}

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

\end{superclasses}

\begin{thebibliography}{10}

\bibitem{Ln19xx}

\end{thebibliography}

\end{document}
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