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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>\begin{align}\bigwedge_{j\in J}\bigvee_{k\in K_j} x_{j,k} =
        \bigvee_{f\in F}\bigwedge_{j\in J} x_{j,f(j)}\end{align}</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. Priestey, ''Introduction to Lattices and Order'' 2nd Edition, Cambridge University Press, 2002, ISBN 0-521-78451-4</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"/>
 
==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]].
 
<!-- This isn't valid. See talk.
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''{{Citation needed|date=February 2007}}:
 
* ''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.
 
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, ''[http://www.springerlink.com/content/nfqh0l29f3unrlwh 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]]

Revision as of 09:05, 9 November 2013

In the mathematical area of order theory, a completely distributive lattice is a complete lattice in which arbitrary joins distribute over arbitrary meets.

Formally, a complete lattice L is said to be completely distributive if, for any doubly indexed family {xj,k | j in J, k in Kj} of L, we have

⋀j∈J⋁k∈Kjxj,k=⋁f∈F⋀j∈Jxj,f(j)

where F is the set of choice functions f choosing for each index j of J some index f(j) in Kj.[1]

Complete distributivity is a self-dual property, i.e. dualizing the above statement yields the same class of complete lattices.[1]

Alternative characterizations

Various different characterizations exist. For example, the following is an equivalent law that avoids the use of choice functionsPotter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park.. 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

⋀{⋁Y∣Y∈S}=⋁{⋀Z∣Z∈S#}

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 LPotter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park.:

  • 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 Lop are continuous posets.

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

Free completely distributive lattices

Every poset C can be 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 ϕ:C→L such that for every completely distributive lattice M and monotonic function f:C→M, there is a unique complete homomorphism fϕ*:L→M satisfying f=fϕ*∘ϕ. For every poset C, the free completely distributive lattice over a poset C exists and is unique up to isomorphism.[2]

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

See also

References

  1. ↑ 1.0 1.1 1.2 B. A. Davey and H. A. Priestey, Introduction to Lattices and Order 2nd Edition, Cambridge University Press, 2002, ISBN 0-521-78451-4
  2. ↑ 2.0 2.1 Joseph M. Morris, Augmenting Types with Unbounded Demonic and Angelic Nondeterminacy, Mathematics of Program Construction, LNCS 3125, 274-288, 2004
  3. ↑ G. N. Raney, Completely distributive complete lattices, Proceedings of the American Mathematical Society, 3: 677 - 680, 1952.
  4. ↑ Alan Hopenwasser, Complete Distributivity, Proceedings of Symposia in Pure Mathematics, 51(1), 285 - 305, 1990.
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