Editing Distributivity (order theory) (section)

==Distributivity laws for complete lattices==
For a [[completeness (order theory)|complete]] lattice, arbitrary subsets have both infima and suprema and thus infinitary meet and join operations are available. Several extended notions of distributivity can thus be described. For example, for the '''infinite distributive law''', finite meets may distribute over arbitrary joins, i.e.

: <math>x \wedge \bigvee S = \bigvee \{ x \wedge s \mid s \in S \}</math>

may hold for all elements ''x'' and all subsets ''S'' of the lattice. Complete lattices with this property are called '''frames''', '''locales''' or '''[[complete Heyting algebra]]s'''. They arise in connection with [[pointless topology]] and [[Stone duality]]. This distributive law ''is not equivalent'' to its dual statement

: <math>x \vee \bigwedge S = \bigwedge \{ x \vee s \mid s \in S \}</math>

which defines the class of dual frames or complete co-Heyting algebras.

Now one can go even further and define orders where arbitrary joins distribute over arbitrary meets. Such structures are called [[completely distributive lattice]]s. However, expressing this requires formulations that are a little more technical. Consider a doubly [[indexed family]] {''x''<sub>''j'',''k''</sub> | ''j'' in ''J'', ''k'' in ''K''(''j'')} of elements of a complete lattice, and let ''F'' be the set of choice functions ''f'' choosing for each index ''j'' of ''J'' some index ''f''(''j'') in ''K''(''j''). A complete lattice is '''completely distributive''' if for all such data the following statement holds:

: <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>

Complete distributivity is again a self-dual property, i.e. dualizing the above statement yields the same class of complete lattices. Completely distributive complete lattices (also called ''completely distributive lattices'' for short) are indeed highly special structures. See the article on [[completely distributive lattice]]s.