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\begin{document}

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\section*{complete Boolean algebra}

\hypertarget{complete_boolean_algebras}{}\section*{{Complete boolean algebras}}\label{complete_boolean_algebras}

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{category_of_complete_boolean_algebras}{Category of complete Boolean algebras}\dotfill \pageref*{category_of_complete_boolean_algebras} \linebreak
\noindent\hyperlink{relation_to_stonean_locales_and_boolean_locales}{Relation to Stonean locales and Boolean locales}\dotfill \pageref*{relation_to_stonean_locales_and_boolean_locales} \linebreak
\noindent\hyperlink{stonean_duality}{Stonean duality}\dotfill \pageref*{stonean_duality} \linebreak
\noindent\hyperlink{cabas}{CABAs}\dotfill \pageref*{cabas} \linebreak
\noindent\hyperlink{algebraicity}{Algebraicity}\dotfill \pageref*{algebraicity} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

A \textbf{complete Boolean algebra} is a [[complete lattice]] that is also a [[Boolean algebra]]. Since lattice homomorphisms of Boolean algebras automatically preserves the Boolean structure, the complete Boolean algebras form a [[full subcategory]] [[CompBoolAlg]] of [[CompLat]].

\hypertarget{category_of_complete_boolean_algebras}{}\subsection*{{Category of complete Boolean algebras}}\label{category_of_complete_boolean_algebras}

A natural notion of [[morphism]] for complete Boolean algebras is that of a continuous homomorphism of Boolean algebras, also known as complete Boolean homomorphisms. These can be defined as homomorphisms of Boolean algebras that preserve [[suprema]], or, equivalently, [[infima]]. It suffices to require preservation of [[suprema]] of [[directed subsets]].

With this notion of morphisms, complete Boolean algebras form a category.

\hypertarget{relation_to_stonean_locales_and_boolean_locales}{}\subsection*{{Relation to Stonean locales and Boolean locales}}\label{relation_to_stonean_locales_and_boolean_locales}

The category of complete Boolean algebras is equivalent to the category of [[Boolean locales]] and [[Stonean locales]]. The latter fact is also known as the (localic) \textbf{Stonean duality}.

In presence of the [[axiom of choice]], the category of [[Stonean locales]] is equivalent to the category of [[Stonean spaces]], so the latter is contravariantly equivalent to the category of complete Boolean algebras. The latter fact is also known as the (traditional) \textbf{Stonean duality}.

\hypertarget{stonean_duality}{}\subsection*{{Stonean duality}}\label{stonean_duality}

The [[Stone duality]] establishes a [[contravariant equivalence]] of [[categories]] between the category of [[Boolean algebras]] and the category of [[Stone spaces]]. The latter is a [[full subcategory]] of the category [[Top]] of [[topological spaces]] and [[continuous maps]] on [[compact]] [[totally disconnected]] [[Hausdorff]] [[topological spaces]].

Recall that a [[Stonean space]] is a [[compact]] [[extremally disconnected]] [[Hausdorff]] [[topological space]]. [[Morphisms]] of [[Stonean spaces]] are defined to be [[open]] [[continuous maps]]. Restricting the [[Stone duality]] produces a [[contravariant equivalence]] between the category of complete Boolean algebres and the category of [[Stonean spaces]]. See Corollary{\tt \symbol{126}}6.10(2) in Bezhanishvili \cite{SDGC}.

\hypertarget{cabas}{}\subsection*{{CABAs}}\label{cabas}

Assuming [[excluded middle]], complete \emph{[[atomic Boolean algebra|atomic]]} Boolean algebras are (up to [[isomorphism]]) precisely [[power sets]]. In fact, taking power sets defines a [[fully faithful functor]] from the [[opposite category]] of [[Set]] to [[Comp Bool Alg]] whose [[essential image]] consists of the complete atomic boolean algebras. See at \emph{\href{Set#OppositeCategory}{Set -- Properties -- Opposite category}}. These abstract representations of power sets are important enough to have their own abbreviation: `CABA'.

This property of CABAs is not applicable in [[constructive mathematics]], where power sets are rarely boolean algebras. However, we can use [[discrete space|discrete]] [[locales]] instead (or rather, their corresponding [[frames]]). That is, define a \textbf{CABA} to be (not a complete atomic boolean algebra but) a frame $X$ such that the locale maps $X \to 1$ and $X \to X \times X$ (which in the category of frames are maps $0 \to X$ and $X + X \to X$) are [[open map|open]] (as locale maps). Then it should be (I will check) a classical theorem that CABAs and complete atomic boolean algebras are the same, and a constructive theorem that CABAs and power sets are the same (in the same functorial manner as above).

\hypertarget{algebraicity}{}\subsection*{{Algebraicity}}\label{algebraicity}

Complete Boolean algebras are the models of an [[algebraic theory]] (in which the operations, notably $j$-indexed [[suprema]] and infima, have arities $j$ unbounded by any cardinal). It follows from general principles that the underlying-set functor $U: CompBoolAlg \to Set$ preserves and reflects limits and isomorphisms.

However, this functor $U$ is \emph{not} [[monadic functor|monadic]]; in fact, it does not even possess a left adjoint. Indeed, while the free complete Boolean algebra on a \emph{finite} set $X$ exists and coincides with the free Boolean algebra on $X$ (it is finite, being isomorphic to the double power set $P(P X)$), we have

\begin{uthm}
There is no free complete Boolean algebra on countably many generators.

\end{uthm}
As a consequence, $CompBoolAlg$ is not cocomplete (otherwise there would exist a countable coproduct of copies of $P(P 1)$, which is ruled out by the previous theorem).

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[Boolean locale]]
\item [[Stonean locale]]

\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

For instance around theorem 2.4 of

\begin{itemize}%
\item [[Jaap van Oosten]], \emph{Basic category theory} (\href{http://www.staff.science.uu.nl/~ooste110/syllabi/catsmoeder.pdf}{pdf})

\end{itemize}
and

\begin{itemize}%
\item PlanetMat, \emph{\href{http://planetmath.org/RepresentingACompleteAtomicBooleanAlgebraByPowerSet.html}{representing a complete atomic Boolean algebra by power set}}

\end{itemize}
The [[Stone duality]] for complete Boolean algebras is explained in

$\backslash$bibitem\{SDGC\} [[Guram Bezhanishvili]], \emph{Stone duality and Gleason covers through de Vries duality}. Topology and its Applications 157 (2010), 1064–1080.

[[!redirects complete Boolean algebra]] [[!redirects complete boolean algebra]] [[!redirects complete Boolean algebras]] [[!redirects complete boolean algebras]] [[!redirects complete Boolean lattice]] [[!redirects complete boolean lattice]] [[!redirects complete Boolean lattices]] [[!redirects complete boolean lattices]] [[!redirects complete Boolean ring]] [[!redirects complete boolean ring]] [[!redirects complete Boolean rings]] [[!redirects complete boolean rings]]

[[!redirects complete atomic Boolean algebra]] [[!redirects complete atomic boolean algebra]] [[!redirects complete atomic Boolean algebras]] [[!redirects complete atomic boolean algebras]] [[!redirects complete atomic Boolean lattice]] [[!redirects complete atomic boolean lattice]] [[!redirects complete atomic Boolean lattices]] [[!redirects complete atomic boolean lattices]] [[!redirects complete atomic Boolean ring]] [[!redirects complete atomic boolean ring]] [[!redirects complete atomic Boolean rings]] [[!redirects complete atomic boolean rings]] [[!redirects caba]] [[!redirects CABA]] [[!redirects cabas]] [[!redirects CABAs]] [[!redirects caba's]] [[!redirects CABA's]] [[!redirects caba's]] [[!redirects CABA's]]



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