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\begin{document}

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\section*{completely distributive lattice}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{category_theory}{}\paragraph*{{$(0,1)$-Category theory}}\label{category_theory}

[[!include (0,1)-category theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{algebraic_lattices}{Algebraic lattices}\dotfill \pageref*{algebraic_lattices} \linebreak
\noindent\hyperlink{completely_distributive_boolean_algebras}{Completely distributive Boolean algebras}\dotfill \pageref*{completely_distributive_boolean_algebras} \linebreak
\noindent\hyperlink{constructive_complete_distributivity}{Constructive complete distributivity}\dotfill \pageref*{constructive_complete_distributivity} \linebreak
\noindent\hyperlink{remark}{Remark}\dotfill \pageref*{remark} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{links}{Links}\dotfill \pageref*{links} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

A \textbf{completely distributive lattice} is a [[complete lattice]] $L$ in which arbitrary [[joins]] and arbitrary meets [[distributive lattice|distribute]] over each other.

More formally: given a complete lattice $L$ and functions $p: J \to I$ and $f: J \to L$, we have

\begin{displaymath}
\bigwedge_{i \in I} \bigvee_{j \in p^{-1}(i)} f(j) \geq \bigvee_{sections\; s: I \to J} \bigwedge_{i \in I} f(s(i))
\end{displaymath}
where ``section'' means [[section]] of $p$. Complete distributivity states that this inequality is an equality, for all $f, p$. The same statement then holds upon switching $\bigwedge$ and $\bigvee$, i.e., complete distributivity is a [[duality|self-dual]] property.

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{algebraic_lattices}{}\subsubsection*{{Algebraic lattices}}\label{algebraic_lattices}

\begin{prop}
\label{}\hypertarget{}{}
The category of [[Alexandroff locales]] is equivalent to that of completely distributive [[algebraic lattice|algebraic lattices]].

\end{prop}
This appears as remark 4.3 in (\hyperlink{Caramello11}{Caramello 2011}).

\hypertarget{completely_distributive_boolean_algebras}{}\subsubsection*{{Completely distributive Boolean algebras}}\label{completely_distributive_boolean_algebras}

\begin{lemma}
\label{}\hypertarget{}{}
A complete totally ordered poset is completely distributive.

\end{lemma}
\begin{proof}
(Note: this uses the [[axiom of choice]].) Suppose we had a strict inequality

\begin{displaymath}
\bigwedge_{i \in I} \bigvee_{j \in p^{-1}(i)} f(j) \gt \bigvee_{sections\; s: I \to J} \bigwedge_{i \in I} f(s(i)).
\end{displaymath}
Denote the left side by $x$ and the right by $y$. Either there is no element $z$ strictly between $x$ and $y$, or there is. In the former case, we have for each $i$ that $\bigvee_{j \in p^{-1}(i)} f(j) \geq x$, and so (using trichotomy) we have $f(j) \gt y$ for some $j \in p^{-1}(i)$. Choosing such a $j$ for each $i$, we obtain a section $s$ with $f(s(i)) \gt y$ for all $y$, whence $f(s(i)) \geq x$ for this case, so that $\bigvee_{sections\; s: I \to J} \bigwedge_{i \in I} f(s(i)) \geq x \gt y$, contradiction. If there is $z$ with $x \gt z \gt y$, we argue similarly to obtain a section $s$ with $f(s(i)) \gt z$ for all $i$, whence $\bigwedge_{i \in I} f(s(i)) \geq z$, and we obtain a contradiction as before with $z$ replacing $x$.

\end{proof}
\begin{prop}
\label{}\hypertarget{}{}
A complete Boolean algebra $B$ is completely distributive iff it is atomic (a [[CABA]]), i.e., is a [[power set]] as a Boolean algebra.

\end{prop}
\begin{proof}
For a complete atomic Boolean algebra $B$, it is classical that the canonical map $B \to P(atoms(B))$, sending each $b \in B$ to the set of [[atoms]] below it, is an isomorphism. Such [[power sets]] are products of copies of $\mathbf{2} = \{0 \leq 1\}$, which is completely distributive by the lemma, and [[products]] of completely distributive lattices are completely distributive.

In the other direction, suppose $B$ is completely distributive. Take $p = \pi_2: \mathbf{2} \times B \to B$, and $\alpha: \mathbf{2} \times B \to B$ by $\alpha(0, b) \coloneqq \neg b$ and $\alpha(1, b) \coloneqq b$. Sections of $p$ correspond to functions $g: B \to \mathbf{2}$, and so complete distributivity gives

\begin{displaymath}
1 = \bigwedge_{b \in B} (b \vee \neg b) = \bigvee_{g: B \to \mathbf{2}} \bigwedge_{b \in B} \alpha(g(b), b).
\end{displaymath}
Write $a(g)$ as abbreviation for $\bigwedge_{b \in B} \alpha(g(b), b)$, we have

\begin{displaymath}
b = b \wedge 1 = b \wedge \bigvee_{g: B \to \mathbf{2}} a(g) = \bigvee_{g: B \to \mathbf{2}} b \wedge a(g)
\end{displaymath}
so $b \wedge a(g) \neq 0$ for some $g$ if $b \neq 0$. Notice that $b \wedge a(g) \neq 0$ implies $g(b) = 1$, from which we infer two things:

\begin{itemize}%
\item Whenever $b \leq a(g)$ with $b \neq 0$, then $a(g) \leq \alpha(g(b), b) = \alpha(1, b) = b$; therefore $a(g)$ is an atom whenever $a(g) \neq 0$;


\item Provided that $b \wedge a(g) \neq 0$, the preceding point gives that $a(g)$ is an atom below $b \wedge a(g) \leq b$.



\end{itemize}
The last point shows every element $b$ has an atom $a(g)$ below it, so that $B$ is atomic, as was to be shown.

\end{proof}
\hypertarget{constructive_complete_distributivity}{}\subsubsection*{{Constructive complete distributivity}}\label{constructive_complete_distributivity}

A complete lattice $A$ is called \textbf{constructively completely distributive} (CCD) if the join-assigning morphism $D A \to A$ has a left adjoint, with $D A$ the poset of [[downsets]].

\emph{Constructive complete distributivity is equivalent to complete distributivity if and only if the [[axiom of choice]] holds} (\hyperlink{WoodFawcett}{Wood\&Fawcett (1990)}).

Constructively completely distributive lattices are an example of [[continuous algebras]] for a [[lax-idempotent 2-monad]].

\hypertarget{remark}{}\subsubsection*{{Remark}}\label{remark}

Completely distributive lattices correspond to tight [[Galois connections]] (Raney 1953). This generalizes to a correspondence between totally distributive toposes and [[essential geometric morphism|essential localizations]] (Lucyshyn-Wright 2011).

CCD lattices are precisely the [[nuclear objects]] in the category of complete lattices.

The (bi-) category $\mathfrak{CCD}$ with CCD lattices and sup-preserving maps is the [[Karoubi envelope|idempotent splitting]] of the (bi-) category of relations $\mathfrak{Rel}$. This plays an important role in \emph{[[domain]]-theoretical} approaches to the semantics of [[linear logic]].

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

\begin{itemize}%
\item [[continuous lattice]]


\item [[totally distributive category]]



\end{itemize}
\hypertarget{links}{}\subsection*{{Links}}\label{links}

\begin{itemize}%
\item wikipedia-entry: \href{http://en.wikipedia.org/wiki/Completely_distributive_lattice}{completely distributive lattice}

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

\begin{itemize}%
\item [[Olivia Caramello]], \emph{A topos-theoretic approach to Stone-type dualities} , ms. 2011. (\href{http://arxiv.org/abs/1103.3493}{arXiv:1103.3493})

\end{itemize}
\begin{itemize}%
\item B. Fawcett, R. J. Wood, \emph{Constructive complete distributivity I} , Math. Proc. Camb. Phil. Soc. \textbf{107} (1990) pp.81-89.

\end{itemize}
\begin{itemize}%
\item R. Guitart, J. Riguet, \emph{Enveloppe Karoubienne de Cat\'e{}gories de Kleisli} , Cah. Top. Geom. Diff. Cat. \textbf{XXXIII} no.3 (1992) pp.261-266. (\href{http://archive.numdam.org/article/CTGDC_1992__33_3_261_0.pdf}{pdf})


\item [[Rory Lucyshyn-Wright|R. Lucyshyn-Wright]], \emph{Totally Distributive Toposes} , arXiv.1108.4032 (2011). (\href{http://arxiv.org/pdf/1108.4032v3}{pdf})


\item G. N. Raney, \emph{Tight Galois Connections and Complete Distributivity} , Trans.Amer.Math.Soc \textbf{97} (1960) pp.418-426. (\href{http://www.ams.org/journals/tran/1960-097-03/S0002-9947-1960-0120171-3/S0002-9947-1960-0120171-3.pdf}{pdf})


\item R. Rosebrugh, R. J. Wood, \emph{Constructive complete distributivity IV} , App. Cat. Struc. \textbf{2} (1994) pp.119-144. (\href{http://www.mta.ca/~rrosebru/articles/ccd4.pdf}{preprint})


\item [[Isar Stubbe|I. Stubbe]], \emph{Towards ``Dynamic Domains'': Totally Continuous Complete Q-Categories} , Theo. Comp. Sci. \textbf{373} no.1-2 (2007) pp.142-160.



\end{itemize}
[[!redirects completely distributive lattices]] [[!redirects constructive completely distributive lattice]] [[!redirects constructively completely distributive lattice]] [[!redirects constructively completely distributive lattices]] [[!redirects constructive completely distributive lattices]] [[!redirects CCD lattice]]



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