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In the mathematical area of [[order theory]], a '''completely distributive lattice''' is a [[complete lattice]] in which arbitrary [[join (lattice theory)|join]]s [[distributivity (order theory)|distribute]] over arbitrary [[meet (lattice theory)|meet]]s.

Formally, a complete lattice ''L'' is said to be '''completely distributive''' if, for any doubly indexed family 
{''x''<sub>''j'',''k''</sub> | ''j'' in ''J'', ''k'' in ''K''<sub>''j''</sub>} of ''L'', we have
: <math>\bigwedge_{j\in J}\bigvee_{k\in K_j} x_{j,k} = 
         \bigvee_{f\in F}\bigwedge_{j\in J} x_{j,f(j)}</math>
where ''F'' is the set of [[choice function]]s ''f'' choosing for each index ''j'' of ''J'' some index ''f''(''j'') in ''K''<sub>''j''</sub>.<ref name="DaveyPriestley">B. A. Davey and H. A. Priestley, ''[[Introduction to Lattices and Order]]'' 2nd Edition, Cambridge University Press, 2002, {{ISBN|0-521-78451-4}}, 10.23 Infinite distributive laws, pp. 239–240</ref>

Complete distributivity is a self-dual property, i.e. [[Duality (order theory)|dualizing]] the above statement yields the same class of complete lattices.<ref name="DaveyPriestley"/>

Without the axiom of choice, no complete lattice with more than one element can ever satisfy the above property, as one can just let ''x''<sub>''j'',''k''</sub> equal the top element of ''L'' for all indices ''j'' and ''k'' with all of the sets ''K''<sub>''j''</sub> being nonempty but having no choice function.{{Citation needed|date=March 2015}}

==Alternative characterizations==

Various different characterizations exist. For example, the following is an equivalent law that avoids the use of choice functions{{Citation needed|date=February 2007}}. For any set ''S'' of sets, we define the set ''S''<sup>#</sup> to be the set of all subsets ''X'' of the complete lattice that have non-empty intersection with all members of ''S''. We then can define complete distributivity via the statement

: <math>\begin{align}\bigwedge \{ \bigvee Y \mid Y\in S\} = \bigvee\{ \bigwedge Z \mid Z\in S^\# \}\end{align}</math>

The operator ( )<sup>#</sup> might be called the '''crosscut operator'''. This version of complete distributivity only implies the original notion when admitting the [[Axiom of Choice]].

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However, the latter version is always equivalent to the statement:

: <math>\begin{align}\bigwedge \{ \bigvee Y \mid Y\in S\} = \bigvee\bigcap S\end{align}</math>

for all sets ''S'' of subsets of a complete lattice.
-->

==Properties==
In addition, it is known that the following statements are equivalent for any complete lattice ''L'':<ref name="Raney53">G. N. Raney, ''[https://www.ams.org/journals/proc/1953-004-04/S0002-9939-1953-0058568-4/S0002-9939-1953-0058568-4.pdf A subdirect-union representation for completely distributive complete lattices]'', Proceedings of the American Mathematical Society, 4: 518 - 522, 1953.</ref>

* ''L'' is completely distributive.
* ''L'' can be embedded into a direct product of chains [0,1] by an [[order embedding]] that preserves arbitrary meets and joins.
* Both ''L'' and its dual order ''L''<sup>op</sup> are [[continuous poset]]s.{{Citation needed|date=January 2016}}

Direct products of [0,1], i.e. sets of all functions from some set ''X'' to [0,1] ordered [[pointwise order|pointwise]], are also called ''cubes''.

==Free completely distributive lattices==<!-- This section is linked from [[Completely distributive lattice]]. See [[WP:MOS#Section management]] -->
Every [[partially ordered set|poset]] ''C'' can be [[Complete lattice#Completion|completed]] in a completely distributive lattice.

A completely distributive lattice ''L'' is called the '''free completely distributive lattice over a poset ''C''''' if and only if there is an [[order embedding]] <math>\phi:C\rightarrow L</math> such that for every completely distributive lattice ''M'' and [[monotonic function]] <math>f:C\rightarrow M</math>, there is a unique [[Complete lattice#Morphisms of complete lattices|complete homomorphism]] <math>f^*_\phi:L\rightarrow M</math> satisfying <math>f=f^*_\phi\circ\phi</math>. For every poset ''C'', the free completely distributive lattice over a poset ''C'' exists and is unique up to isomorphism.<ref name="Morris04">Joseph M. Morris, ''[https://doi.org/10.1007%2F978-3-540-27764-4_15 Augmenting Types with Unbounded Demonic and Angelic Nondeterminacy]'', Mathematics of Program Construction, LNCS 3125, 274-288, 2004</ref>

This is an instance of the concept of [[free object]]. Since a set ''X'' can be considered as a poset with the discrete order, the above result guarantees the existence of the free completely distributive lattice over the set ''X''.

==Examples==
* The [[unit interval]] [0,1], ordered in the natural way, is a completely distributive lattice.<ref name="Raney52">G. N. Raney, ''Completely distributive complete lattices'', Proceedings of the [[American Mathematical Society]], 3: 677 - 680, 1952.</ref>
**More generally, any [[Total order#Completeness|complete chain]] is a completely distributive lattice.<ref name="hopenwasser90">Alan Hopenwasser, ''Complete Distributivity'', Proceedings of Symposia in Pure Mathematics, 51(1), 285 - 305, 1990.</ref>
* The [[power set]] lattice <math>(\mathcal{P}(X),\subseteq)</math> for any set ''X'' is a completely distributive lattice.<ref name="DaveyPriestley"/>
* For every poset ''C'', there is a ''free completely distributive lattice over C''.<ref name="Morris04"/> See the section on [[Completely distributive lattice#Free completely distributive lattices|Free completely distributive lattices]] above.

==See also==
* [[Glossary of order theory]]
* [[Distributive lattice]]

==References==

<references/>

[[Category:Order theory]]