Mathematical Structures: Boolean algebras with operators

Boolean algebras with operators

http://mathcs.chapman.edu/structuresold/files/Boolean_algebras_with_operators.pdf
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\begin{document}
\textbf{\Large Boolean algebras with operators}

\abbreviation{BAO}

\begin{definition}
A \emph{Boolean algebra with operators} is a structure $\mathbf{A}=\langle A,\vee,0, \wedge,1,\neg,f_i\ (i\in I)\rangle$ such that

$\langle A,\vee,0,\wedge,1,\neg\rangle$ is a Boolean algebra

$f_i$ is \emph{join-preserving} in each argument:
$f_i(\ldots,x\vee y,\ldots)=f_i(\ldots,x,\ldots)\vee f_i(\ldots,y,\ldots)$

$f_i$ is \emph{normal} in each argument:  $f_i(\ldots,0,\ldots)=0$
\end{definition}

\begin{morphisms}
Let $\mathbf{A}$ and $\mathbf{B}$ be Boolean algebras with operators of the same signature.
A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a Boolean homomorphism and preserves all the operators:

$h(f_i(x_0,\ldots,x_{n-1}))=f_i(h(x_0),\ldots,h(x_{n-1}))$
\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
Equational theory & decidable\\\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 & yes, $n=2$\\\hline
Congruence regular & yes\\\hline
Congruence uniform & yes\\\hline
Congruence extension property & yes\\\hline
Definable principal congruences & no\\\hline
Equationally def. pr. cong. & no\\\hline
Amalgamation property & yes\\\hline
Strong amalgamation property & yes\\\hline
Epimorphisms are surjective & yes\\\hline
\end{tabular}
\end{properties}
\end{table}

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\begin{subclasses}\

\href{Modal_algebras.pdf}{Modal algebras}

\href{Boolean_monoids.pdf}{Boolean monoids}

\end{subclasses}

\begin{superclasses}\

\href{Boolean_algebras.pdf}{Boolean algebras}

\end{superclasses}

\begin{thebibliography}{10}

\bibitem{Ln19xx}

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\end{document}
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