Boolean algebras

http://math.chapman.edu/structuresold/files/Boolean_algebras.pdf

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\begin{document}
\textbf{\Large Boolean algebras}
\quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Boolean_algebras}{edit}

\abbreviation{BA}

\begin{definition}
A \emph{Boolean algebra} is a structure $\mathbf{A}=\left\langle A,\vee
,0,\wedge ,1,-\right\rangle $ of type $\left\langle 2,0,2,0,1\right\rangle $
such that

$0,1$ are identities for $\vee,\wedge$:  $x\vee 0=x$, $x\wedge 1=x$

$-$ gives a complement:  $x\wedge -x=0$, $x\vee -x=1$

$\vee,\wedge$ are associative:  $x\vee (y\vee z)=(x\vee y)\vee z$, $x\wedge (y\wedge z)=(x\wedge y)\wedge z$

$\vee,\wedge$ are commutative:  $x\vee y=y\vee x$, $x\wedge y=y\wedge x$

$\vee,\wedge$ are mutually distributive:  $x\wedge (y\vee z)=(x\wedge y)\vee (x\wedge z)$, $x\vee (y\wedge z)=(x\vee y)\wedge (x\vee z)$
\end{definition}

\begin{definition}
A \emph{Boolean algebra} is a structure $\mathbf{A}=\left\langle A,\vee
,0,\wedge ,1,-\right\rangle $ of type $\left\langle 2,0,2,0,1\right\rangle $
such that

$\left\langle A,\vee ,0,\wedge ,1\right\rangle $ is a 
\href{Bounded_distributive_lattices.pdf}{bounded distributive lattice}

$-$ gives a complement:  $x\wedge -x=0$, $x\vee -x=1$
\end{definition}

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

$h(x\vee y)=h(x)\vee h(y)$, $h(-x)=-h(x)$

It follows that $h(x\wedge y)=h(x)\wedge h(y)$, $h(0)=0$, $h(1)=1$.
\end{morphisms}

\begin{definition}
A \emph{Boolean ring} is a structure $\mathbf{A}=\left\langle A,+
,0,\cdot ,1\right\rangle $ of type $\left\langle 2,0,2,0\right\rangle $
such that

$\left\langle A,+ ,0,\cdot ,1\right\rangle $ is a \href{Commutative_rings_with_unit.pdf}{commutative ring with unit}

$\cdot$ is idempotent:  $x\cdot x=x$

Remark: 
The term-equivalence with Boolean algebras is given by $x\wedge y=x\cdot y$, $-x=x+1$, $x\vee y=-(-x\wedge -y)$ and 
$x+y=(x\vee y)\wedge -(x\wedge y)$.
\end{definition}

\begin{definition}
A \emph{Boolean algebra} is a \href{Heyting_algebras.pdf}{Heyting algebra} $\mathbf{A}=\left\langle
A,\vee ,0,\wedge ,1,\rightarrow \right\rangle $ such that

$x\rightarrow 0$ is an involution:  $\left( x\rightarrow 0\right) \rightarrow 0=x$
\end{definition}

\begin{basic_results}
\end{basic_results}

\begin{examples}
\begin{example}
$\langle \mathcal P(S), \cup ,\emptyset, \cap, S, -\rangle$, the
collection of subsets of a sets $S$, with union, intersection, and
setcomplementation.
\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 in NPTIME\\\hline
Quasiequational theory & decidable\\\hline
First-order theory & decidable\\\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 & yes\\\hline
Equationally def. pr. cong. & yes\\\hline
Amalgamation property & yes\\\hline
Strong amalgamation property & yes\\\hline
Epimorphisms are surjective & yes\\\hline
Locally finite & yes\\\hline
Residual size & 2\\\hline
\end{tabular}
\end{properties}
\end{table}

\begin{finite_members} 
Number of algebras $=\left\{ 
\begin{array}{cc}
1 & \text{if size}=2^{n} \\ 
0 & \text{otherwise}\end{array}\right. $
\end{finite_members}

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

\href{One-element_algebras.pdf}{One-element algebras} 

\href{Complete_Boolean_algebras.pdf}{Complete Boolean algebras} 

\end{subclasses}
\begin{superclasses}\ 

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

\href{Generalized_Boolean_algebras.pdf}{Generalized Boolean algebras} 

\href{Heyting_algebras.pdf}{Heyting algebras} 

\end{superclasses}

\begin{thebibliography}{10}

\bibitem{Ln19xx}

\end{thebibliography}

\end{document}
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