Mathematical Structures: Boolean semilattices

# Boolean semilattices

http://mathcs.chapman.edu/structuresold/files/Boolean_semilattices.pdf
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\begin{document}
\textbf{\Large Boolean semilattices}

\abbreviation{BSlat}

\begin{definition}
A \emph{Boolean semilattice} is a structure $\mathbf{A}=\langle A,\vee,0, \wedge,1,\neg,\cdot\rangle$ such that

$\mathbf{A}$ is in the variety generated by complex algebras of semilattices

Let $\mathbf{S}=\langle S,\cdot\rangle$ be a \href{Semilattices.pdf}{semilattice}. The
\emph{complex algebra} of $\mathbf{S}$ is
$Cm(\mathbf{S})=\langle P(S),\cup,\emptyset,\cap,S,-,\cdot\rangle$,
where $\langle P(S),\cup,\emptyset, \cap,S,-\rangle$ is the Boolean algebra of subsets of $S$, and

$X\cdot Y=\{x\cdot y\mid x\in X,\ y\in Y\}$.
\end{definition}

\begin{morphisms}
Let $\mathbf{A}$ and $\mathbf{B}$ be Boolean semilattices.
A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a Boolean homomorphism and preserves $\cdot$:

$h(x\cdot y)=h(x)\cdot h(y)$
\end{morphisms}

\begin{basic_results}
\end{basic_results}

\begin{examples}
\begin{example}
\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
Finitely axiomatizable & open\\\hline
Equational theory & \\\hline
Quasiequational theory & \\\hline
First-order theory & \\\hline
Locally finite & no\\\hline
Residual size & unbounded\\\hline
Congruence distributive & yes\\\hline
Congruence modular & yes\\\hline
Congruence n-permutable & yes, $n=2$\\\hline
Congruence regular & yes\\\hline
Congruence uniform & \\\hline
Congruence extension property & yes\\\hline
Definable principal congruences & \\\hline
Equationally def. pr. cong. & \\\hline
Amalgamation property & \\\hline
Strong amalgamation property & \\\hline
Epimorphisms are surjective & \\\hline
\end{tabular}
\end{properties}
\end{table}

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

$\begin{array}{lr} f(1)= &1\\ f(2)= &1\\ f(3)= &0\\ f(4)= &5\\ f(5)= &0\\ f(6)= &0\\ f(7)= &0\\ f(8)= &\ge 97\text{ out of }104\\ \end{array}$
\end{finite_members}

\href{http://math.chapman.edu/cgi-bin/structures?Some_members_of_BSlat}{Some members of BSlat}
\hyperbaseurl{http://math.chapman.edu/structures/files/}
\begin{subclasses}\

\href{Variety_generated_by_complex_algebras_of_linear_semilattices.pdf}{Variety generated by complex algebras of linear semilattices}

\end{subclasses}
\begin{superclasses}\

\href{Commutative_Boolean_semigroups.pdf}{Commutative Boolean semigroups}

\end{superclasses}

\begin{thebibliography}{10}

\bibitem{Ln19xx}

\end{thebibliography}

\end{document}
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