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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.
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| Formally, a complete lattice ''L'' is said to be '''completely distributive''' if, for any doubly indexed family
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| {''x''<sub>''j'',''k''</sub> | ''j'' in ''J'', ''k'' in ''K''<sub>''j''</sub>} of ''L'', we have
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| : <math>\begin{align}\bigwedge_{j\in J}\bigvee_{k\in K_j} x_{j,k} =
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| \bigvee_{f\in F}\bigwedge_{j\in J} x_{j,f(j)}\end{align}</math>
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| 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. Priestey, ''Introduction to Lattices and Order'' 2nd Edition, Cambridge University Press, 2002, ISBN 0-521-78451-4</ref>
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| 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"/>
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| ==Alternative characterizations==
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| 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
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| : <math>\begin{align}\bigwedge \{ \bigvee Y \mid Y\in S\} = \bigvee\{ \bigwedge Z \mid Z\in S^\# \}\end{align}</math>
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| 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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| <!-- This isn't valid. See talk.
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| However, the latter version is always equivalent to the statement:
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| : <math>\begin{align}\bigwedge \{ \bigvee Y \mid Y\in S\} = \bigvee\bigcap S\end{align}</math>
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| for all sets ''S'' of subsets of a complete lattice.
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| -->
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| ==Properties==
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| In addition, it is known that the following statements are equivalent for any complete lattice ''L''{{Citation needed|date=February 2007}}:
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| * ''L'' is completely distributive.
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| * ''L'' can be embedded into a direct product of chains [0,1] by an [[order embedding]] that preserves arbitrary meets and joins.
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| * Both ''L'' and its dual order ''L''<sup>op</sup> are [[continuous poset]]s.
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| 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''.
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| ==Free completely distributive lattices==<!-- This section is linked from [[Completely distributive lattice]]. See [[WP:MOS#Section management]] -->
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| Every [[partially ordered set|poset]] ''C'' can be [[Complete lattice#Completion|completed]] in a completely distributive lattice.
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| 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, ''[http://www.springerlink.com/content/nfqh0l29f3unrlwh Augmenting Types with Unbounded Demonic and Angelic Nondeterminacy]'', Mathematics of Program Construction, LNCS 3125, 274-288, 2004</ref>
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| 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''.
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| ==Examples==
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| * 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>
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| **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>
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| * The [[power set]] lattice <math>(\mathcal{P}(X),\subseteq)</math> for any set ''X'' is a completely distributive lattice.<ref name="DaveyPriestley"/>
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| * 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.
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| ==See also==
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| * [[Glossary of order theory]]
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| * [[Distributive lattice]]
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| ==References==
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| <references/>
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| [[Category:Order theory]]
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Nice to meet you, my name is Refugia. I am a meter reader. What I adore performing is performing ceramics but I haven't made a dime with it. South Dakota is where me and my husband live and my family members loves it.
Here is my blog post :: std testing at home