History of Boolean monoids

 
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 \begin{document}
 \textbf{\Large Boolean monoids}
 \quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Boolean_monoids}{edit}
 
 \abbreviation{BMon}
 \begin{definition}
 A \emph{Boolean monoid} is a structure $\mathbf{A}=\langle A,\vee,0,
 \wedge,1,\neg,\cdot,e\rangle$ such that
 
 
 $\langle A,\vee,0,
 \wedge,1,\neg\rangle $ is a \href{Boolean_algebras.pdf}{Boolean algebra}
 
 
 $\langle A,\cdot,e\rangle $ is a \href{Monoids.pdf}{monoids}
 
 
 $\cdot$ is \emph{join-preserving} in each argument:  
 $(x\vee y)\cdot z=(x\cdot z)\vee (y\cdot z) \mbox{ and } x\cdot (y\vee z)=(x\cdot y)\vee (x\cdot z)$
 
 
 $\cdot$ is \emph{normal} in each argument:  $0\cdot x=0 \mbox{ and } x\cdot 0=0$
 
 
 Remark: 
 
 \end{definition}
 \begin{morphisms}
 Let $\mathbf{A}$ and $\mathbf{B}$ be Boolean monoids. 
 A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a Boolean homomorphism and preserves $\cdot$, $e$:
 
 $h(x\cdot y)=h(x)\cdot h(y) \mbox{ and } h(e)=e$
 \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 & \\\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 & yes\\\hline
 Congruence extension property & yes\\\hline
 Definable principal congruences & no\\\hline
 Equationally def. pr. cong. & no\\\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 $n$.
 
 $\begin{array}{lr}
 f(1)= &1\\
 f(2)= &1\\
 f(3)= &0\\
 f(4)= &9\\
 f(5)= &0\\
 f(6)= &0\\
 f(7)= &0\\
 f(8)= &258\\
 \end{array}$
 \end{finite_members}
 \hyperbaseurl{http://math.chapman.edu/structures/files/}
 \parskip0pt
 \begin{subclasses}\ 
 
 \href{Sequential_algebras.pdf}{Sequential algebras} 
 
 \end{subclasses}
 \begin{superclasses}\ 
 
 \href{Boolean_algebras_with_operators.pdf}{Boolean algebras with operators} 
 
 \end{superclasses}
 
 \begin{thebibliography}{10}
 
 \bibitem{Ln19xx}
 
 \end{thebibliography}
 
 \end{document}
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