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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''''j'',''k'' | ''j'' in ''J'', ''k'' in ''K''''j''} 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''''j''.<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''''j'',''k'' equal the top element of ''L'' for all indices ''j'' and ''k'' with all of the sets ''K''''j'' 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''# 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 ( )# might be called the '''crosscut operator'''. This version of complete distributivity only implies the original notion when admitting the [[Axiom of Choice]].



==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''op 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==
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]]

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wikipedia>David Eppstein: link Introduction to Lattices and Order and add pages