Editing Distributivity (order theory)

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In the [[mathematics|mathematical]] area of [[order theory]], there are various notions of the common concept of [[distributivity]], applied to the formation of [[supremum|suprema]] and [[infimum|infima]]. Most of these apply to [[partially ordered set]]s that are at least [[lattice (order)|lattices]], but the concept can in fact reasonably be generalized to [[semilattice]]s as well.

==Distributive lattices==
Probably the most common type of distributivity is the one defined for [[lattice (order)|lattices]], where the formation of binary suprema and infima provide the total operations of join (<math>\vee</math>) and meet (<math>\wedge</math>). Distributivity of these two operations is then expressed by requiring that the identity

: <math>x \wedge (y \vee z) = (x \wedge y) \vee (x \wedge z)</math>

hold for all elements ''x'', ''y'', and ''z''. This distributivity law defines the class of '''[[distributive lattice]]s'''. Note that this requirement can be rephrased by saying that binary meets preserve binary joins. The above statement is known to be equivalent to its [[duality (order theory)|order dual]]

: <math>x \vee (y \wedge z) = (x \vee y) \wedge (x \vee z)</math>

such that one of these properties suffices to define distributivity for lattices. Typical examples of distributive lattice are [[totally ordered set]]s, [[Boolean algebra (structure)|Boolean algebra]]s, and [[Heyting algebra]]s. Every finite distributive lattice is [[Order isomorphism|isomorphic]] to a lattice of sets, ordered by inclusion ([[Birkhoff's representation theorem]]).

==Distributivity for semilattices==
[[File:DistrSemilattice.svg|thumb|Hasse diagram for the definition of distributivity for a meet-semilattice.]]
A [[semilattice]] is [[partially ordered set]] with only one of the two lattice operations, either a '''meet-''' or a '''join-semilattice'''. Given that there is only one [[binary operation]], distributivity obviously cannot be defined in the standard way. Nevertheless, because of the interaction of the single operation with the given order, the following definition of distributivity remains possible. A '''meet-semilattice''' is '''distributive''', if for all ''a'', ''b'', and ''x'':

: If ''a'' ∧ ''b'' ≤ ''x'' then there exist ''a''{{′}} and ''b''{{′}} such that ''a'' ≤ ''a''{{′}}, ''b'' ≤ ''b' '' and ''x'' = ''a''{{′}} ∧ ''b' ''.

Distributive join-semilattices are defined [[duality (order theory)|dually]]: a '''join-semilattice''' is '''distributive''', if for all ''a'', ''b'', and ''x'':

: If ''x'' &le; ''a'' &or; ''b'' then there exist ''a''{{′}} and ''b''{{′}} such that ''a''{{′}} ≤ ''a'', ''b''{{′}} ≤ ''b'' and ''x'' = ''a''{{′}} ∨ ''b' ''.

In either case, a' and b' need not be unique.
These definitions are justified by the fact that given any lattice ''L'', the following statements are all equivalent:

* ''L'' is distributive as a meet-semilattice
* ''L'' is distributive as a join-semilattice
* ''L'' is a distributive lattice.

Thus any distributive meet-semilattice in which binary joins exist is a distributive lattice. 
A join-semilattice is distributive [[if and only if]] the lattice of its [[ideal (order theory)|ideals]] (under inclusion) is distributive.<ref>{{cite book| author=G. Grätzer| title=Lattice Theory: Foundation| year=2011| publisher=Springer/Birkhäuser}}; here: Sect. II.5.1, p.167</ref>

This definition of distributivity allows generalizing some statements about distributive lattices to distributive semilattices.

==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.

==Distributive elements in arbitrary lattices==
[[File:N5 1xyz0.svg|thumb|150px|Pentagon lattice ''N''<sub>5</sub>]]

In an arbitrary lattice, an element ''x'' is called a ''distributive element'' if ∀''y'',''z'': {{nowrap|1= ''x'' ∨ (''y'' ∧ ''z'') }} = {{nowrap|1= (''x'' ∨ ''y'') ∧ (''x'' ∨ ''z''). }}
An element ''x'' is called a ''dual distributive element'' if ∀''y'',''z'': {{nowrap|1= ''x'' ∧ (''y'' ∨ ''z'') }} = {{nowrap|1= (''x'' ∧ ''y'') ∨ (''x'' ∧ ''z''). }}

In a distributive lattice, every element is of course both distributive and dual distributive.
In a non-distributive lattice, there may be elements that are distributive, but not dual distributive (and vice versa).
For example, in the depicted pentagon lattice ''N''<sub>5</sub>, the element ''x'' is distributive,<ref>{{cite book | isbn=3-7643-6996-5 | author=George Grätzer | title=General Lattice Theory | location=Basel | publisher=Birkhäuser | edition=2nd | year=2003 }} Here: Def. III.2.1 and the subsequent remark, p.181.</ref> but not dual distributive, since {{nowrap|1= ''x'' ∧ (''y'' ∨ ''z'') }} = {{nowrap|1= ''x'' ∧ 1 }} = ''x'' ≠ ''z'' = {{nowrap|1= 0 ∨ ''z'' }} = {{nowrap|1= (''x'' ∧ ''y'') ∨ (''x'' ∧ ''z''). }}

In an arbitrary lattice ''L'', the following are equivalent:
* ''x'' is a distributive element;
* The map φ defined by φ(''y'') = ''x'' ∨ ''y'' is a [[lattice homomorphism]] from ''L'' to the [[Upper_set#Upper_closure_and_lower_closure|upper closure]] ↑''x'' = { ''y'' ∈ ''L'': ''x'' ≤ ''y'' };
* The [[binary relation]] Θ<sub>''x''</sub> on ''L'' defined by ''y'' Θ<sub>''x''</sub> ''z'' if ''x'' ∨ ''y'' = ''x'' ∨ ''z'' is a [[congruence relation]], that is, an [[equivalence relation]] compatible with ∧ and ∨.<ref>Grätzer (2003), Thm.III.2.2 [originally by [[Øystein Ore|O. Ore]] 1935], p.181-182.</ref>

In an arbitrary lattice, if ''x''<sub>1</sub> and ''x''<sub>2</sub> are distributive elements, then so is ''x''<sub>1</sub> ∨ ''x''<sub>2</sub>.<ref>Grätzer (2003), Thm.III.2.9.(i), p.188</ref>

==Literature==
''Distributivity is a basic concept that is treated in any textbook on lattice and order theory. See the literature given for the articles on [[order theory]] and [[lattice theory]]. More specific literature includes:''

* G. N. Raney, ''Completely distributive complete lattices'', Proceedings of the [[American Mathematical Society]], 3: 677 - 680, 1952.
==References==
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[[Category:Order theory]]