Editing Distributive lattice

{{Short description|Special type of lattice}}
{{More citations needed|date=May 2011}}
In [[mathematics]],  a '''distributive lattice''' is a [[lattice (order)|lattice]] in which the operations of [[join and meet]] [[distributivity|distribute]] over each other. The prototypical examples of such structures are collections of sets for which the lattice operations can be given by set [[union (set theory)|union]] and [[intersection (set theory)|intersection]]. Indeed, these lattices of sets describe the scenery completely: every distributive lattice is&mdash;up to [[order isomorphism|isomorphism]]&mdash;given as such a lattice of sets.

==Definition==
As in the case of arbitrary lattices, one can choose to consider a distributive lattice ''L'' either as a structure of [[order theory]] or of [[universal algebra]]. Both views and their mutual correspondence are discussed in the article on [[lattice (order)|lattices]]. In the present situation, the algebraic description appears to be more convenient.

A lattice (''L'',∨,∧) is '''distributive''' if the following additional identity holds for all ''x'', ''y'', and ''z'' in ''L'':

: ''x'' ∧ (''y'' ∨ ''z'') = (''x'' ∧ ''y'') ∨ (''x'' ∧ ''z'').

Viewing lattices as partially ordered sets, this says that the meet operation preserves non-empty finite joins. It is a basic fact of lattice theory that the above condition is equivalent to its [[duality (order theory)|dual]]:<ref>{{cite book| last=Birkhoff | first=Garrett | authorlink = Garrett Birkhoff|title=Lattice Theory | url=https://archive.org/details/latticetheory0000birk | url-access=registration | year=1967 | edition=3rd | publisher=[[American Mathematical Society]] | series=Colloquium Publications | isbn=0-8218-1025-1 | page=[https://archive.org/details/latticetheory0000birk/page/11 11]}} §6, Theorem 9</ref>

: ''x'' ∨ (''y'' ∧ ''z'') = (''x'' ∨ ''y'') ∧ (''x'' ∨ ''z'') &nbsp; for all ''x'', ''y'', and ''z'' in ''L''.

In every lattice, if one defines the order relation ''p''≤''q'' as usual to mean ''p''∧''q''=''p'', then the inequality ''x'' ∧ (''y'' ∨ ''z'') ≥ (''x'' ∧ ''y'') ∨ (''x'' ∧ ''z'') and its dual ''x'' ∨ (''y'' ∧ ''z'') ≤ (''x'' ∨ ''y'') ∧ (''x'' ∨ ''z'') are always true. A lattice is distributive if one of the converse inequalities holds, too.
More information on the relationship of this condition to other distributivity conditions of order theory can be found in the article [[Distributivity (order theory)]].

== Morphisms ==

A morphism of distributive lattices is just a lattice homomorphism as given in the article on [[lattice (order)|lattices]], i.e. a function that is compatible with the two lattice operations. Because such a morphism of lattices preserves the lattice structure, it will consequently also preserve the distributivity (and thus be a morphism of distributive lattices).

== Examples ==
[[File:Young's lattice.svg|thumb|[[Young's lattice]]]]
Distributive lattices are ubiquitous but also rather specific structures. As already mentioned the main example for distributive lattices are lattices of sets, where join and meet are given by the usual set-theoretic operations. Further examples include:

* The [[Lindenbaum–Tarski algebra|Lindenbaum algebra]] of most [[logic]]s that support [[logical conjunction|conjunction]] and [[Logical disjunction|disjunction]] is a distributive lattice, i.e. "and" distributes over "or" and vice versa.
* Every [[Boolean algebra (structure)|Boolean algebra]] is a distributive lattice.
* Every [[Heyting algebra]] is a distributive lattice. Especially this includes all [[complete Heyting algebra|locales]] and hence all [[open set]] lattices of [[topological space]]s. Also note that Heyting algebras can be viewed as Lindenbaum algebras of [[intuitionistic logic]], which makes them a special case of the first example.
* Every [[Total order|totally ordered set]] is a distributive lattice with max as join and min as meet.
* The [[natural number]]s form a (conditionally complete) distributive lattice by taking the [[greatest common divisor]] as meet and the [[least common multiple]] as join. This lattice also has a least element, namely 1, which therefore serves as the identity element for joins.
* Given a positive integer ''n'', the set of all positive [[divisor]]s of ''n'' forms a distributive lattice, again with the greatest common divisor as meet and the least common multiple as join. This is a Boolean algebra [[if and only if]] ''n'' is [[Square-free integer|square-free]].
* A [[Riesz space|lattice-ordered vector space]] is a distributive lattice.
*[[Young's lattice]] given by the inclusion ordering of [[Young diagram#Diagrams|Young diagrams]] representing [[integer partition]]s is a distributive lattice.
* The points of a [[distributive polytope]] (a [[convex polytope]] closed under coordinatewise minimum and coordinatewise maximum operations), with these two operations as the join and meet operations of the lattice.<ref>{{citation
 | last1 = Felsner | first1 = Stefan
 | last2 = Knauer | first2 = Kolja
 | doi = 10.1016/j.ejc.2010.07.011
 | issue = 1
 | journal = [[European Journal of Combinatorics]]
 | mr = 2727459
 | pages = 45–59
 | title = Distributive lattices, polyhedra, and generalized flows
 | volume = 32
 | year = 2011| doi-access = free
 }}.</ref>

Early in the development of the lattice theory [[Charles S. Peirce]] believed that all lattices are distributive, that is, distributivity follows from the rest of the lattice axioms.<ref name="Fisch.Kloesel.Peirce.1989">{{citation
 | first1 = Charles S. | last1 = Peirce | authorlink1 = Charles Sanders Peirce
 | first2 = M. H. | last2 = Fisch
 | first3 = C. J. W. | last3 = Kloesel
 | title = Writings of Charles S. Peirce: 1879–1884
 | url = https://books.google.com/books?id=E7ZUnx3FqrcC
 | year = 1989
 | publisher=Indiana University Press| isbn = 0-253-37204-6 }}, p. xlvii.</ref><ref>{{cite journal | author=Charles S. Peirce | title=On the Algebra of Logic | journal=[[American Journal of Mathematics]] | volume=3 | pages=15–57 | jstor=2369442 | year=1880 | issue=1 | doi=10.2307/2369442}}, p. 33 bottom</ref>
However, independence [[mathematical proof|proofs]] were given by [[Ernst Schröder (mathematician)|Schröder]], Voigt,<sup>([[:de:Andreas Heinrich Voigt|de]])</sup> [[Jacob Lüroth|Lüroth]], [[Alwin Korselt|Korselt]],<ref>{{cite journal | author=A. Korselt | title=Bemerkung zur Algebra der Logik | journal=[[Mathematische Annalen]]| volume=44 | pages=156–157 | url=http://gdz-lucene.tc.sub.uni-goettingen.de/gcs/gcs?&&action=pdf&metsFile=PPN235181684_0044&divID=LOG_0017&pagesize=original&pdfTitlePage=http://gdz.sub.uni-goettingen.de/dms/load/pdftitle/?metsFile=PPN235181684_0044%7C&targetFileName=PPN235181684_0044_LOG_0017.pdf& | year=1894 | doi=10.1007/bf01446978}} Korselt's non-distributive lattice example is a variant of ''M''<sub>3</sub>, with 0, 1, and ''x'', ''y'', ''z'' corresponding to the empty set, a [[Line (geometry)|line]], and three distinct points on it, respectively.</ref> and [[Richard Dedekind|Dedekind]].<ref name="Fisch.Kloesel.Peirce.1989"/>

== Characteristic properties ==
{{multiple image
 | width = 150
 | footer = [[Hasse diagram]]s of the two prototypical non-distributive lattices.  The diamond lattice ''M''<sub>3</sub> is non-distributive because {{nowrap|''x'' ∧ (''y'' ∨ ''z'')}} = ''x'' ∧ 1 = ''x'' ≠ 0 = 0 ∨ 0 = {{nowrap|(''x'' ∧ ''y'') ∨ (''x'' ∧ ''z'')}}, while the pentagon lattice ''N''<sub>5</sub> is non-distributive because {{nowrap|''x'' ∧ (''y'' ∨ ''z'')}} = ''x'' ∧ 1 = ''x'' ≠ ''z'' = 0 ∨ ''z'' = {{nowrap|(''x'' ∧ ''y'') ∨ (''x'' ∧ ''z'')}}
 | image1 = M3 1xyz0.svg
 | caption1 = diamond lattice ''M''<sub>3</sub>
 | image2 = N5 1xyz0.svg
 | caption2 = pentagon lattice ''N''<sub>5</sub>
}}
Various equivalent formulations to the above definition exist. For example, ''L'' is distributive [[if and only if]] the following holds for all elements ''x'', ''y'', ''z'' in ''L'':
<math display="block"> (x \wedge y) \vee (y \wedge z) \vee (z \wedge x) = (x \vee y) \wedge (y \vee z) \wedge (z \vee x).</math>
Similarly, ''L'' is distributive if and only if

: <math>x \wedge z = y \wedge z</math> and <math>x \vee z = y \vee z</math> always imply <math>x = y.</math>

[[File:Non-dstrbtive lattices-warning.png|thumb|Distributive lattice which contains N5 (solid lines, left) and M3 (right) as sub''set'', but not as sub''lattice'']]
The simplest ''non-distributive'' lattices are ''M''<sub>3</sub>, the "diamond lattice", and ''N''<sub>5</sub>, the "pentagon lattice". A lattice is distributive if and only if none of its sublattices is isomorphic to ''M''<sub>3</sub> or&nbsp;''N''<sub>5</sub>; a sublattice is a subset that is closed under the meet and join operations of the original lattice. Note that this is not the same as being a subset that is a lattice under the original order (but possibly with different join and meet operations). Further characterizations derive from the representation theory in the next section.

An alternative way of stating the same fact is that every distributive lattice is a [[subdirect product]] of copies of the [[Two-element Boolean algebra|two-element chain]], or that the only [[Subdirectly irreducible algebra|subdirectly irreducible]] member of the class of distributive lattices is the two-element chain. As a corollary, every [[Boolean algebra (structure)|Boolean lattice]] has this property as well.<ref>Balbes and Dwinger (1975), p. 63 citing Birkhoff, G. "Subdirect unions in universal algebra", [[Bulletin of the American Mathematical Society]] SO (1944), 764-768.</ref>

Finally distributivity entails several other pleasant properties. For example, an element of a distributive lattice is [[Lattice (order)#Important lattice-theoretic notions|meet-prime]] if and only if it is [[Lattice (order)#Important lattice-theoretic notions|meet-irreducible]], though the latter is in general a weaker property. By duality, the same is true for [[Lattice (order)#Important lattice-theoretic notions|join-prime]] and [[Lattice (order)#Important lattice-theoretic notions|join-irreducible]] elements.<ref>See [[Birkhoff's representation theorem#The partial order of join-irreducibles]].</ref> If a lattice is distributive, its [[covering relation]] forms a [[median graph]].<ref>{{citation
 | first1 = Garrett | last1 = Birkhoff | authorlink1 = Garrett Birkhoff
 | first2 = S. A. | last2 = Kiss
 | title = A ternary operation in distributive lattices
 | journal = Bulletin of the American Mathematical Society
 | volume = 53 | issue = 1 | year = 1947 | pages = 749–752
 | mr = 0021540 
 | url = http://projecteuclid.org/euclid.bams/1183510977
 | doi = 10.1090/S0002-9904-1947-08864-9| doi-access = free}}.</ref>

Furthermore, every distributive lattice is also [[modular lattice|modular]].

== Representation theory ==

The introduction already hinted at the most important characterization for distributive lattices: a lattice is distributive if and only if it is isomorphic to a lattice of sets (closed under [[Union (set theory)|set union]] and [[Intersection (set theory)|intersection]]). (The latter structure is sometimes called a [[ring of sets]] in this context.) That set union and intersection are indeed distributive in the above sense is an elementary fact. The other direction is less trivial, in that it requires the [[representation theorem]]s stated below. The important insight from this characterization is that the identities (equations) that hold in all distributive lattices are exactly the ones that hold in all lattices of sets in the above sense.

[[Birkhoff's representation theorem]] for distributive lattices states that every ''finite'' distributive lattice is isomorphic to the lattice of [[Upper set|lower set]]s of the [[Partially ordered set|poset]] of its join-prime (equivalently: join-irreducible) elements. This establishes a [[bijection]] (up to [[isomorphism]]) between the class of all finite posets and the class of all finite distributive lattices. This bijection can be extended to a [[equivalence of categories|duality of categories]] between homomorphisms of finite distributive lattices and [[Monotonic function|monotone function]]s of finite posets. Generalizing this result to infinite lattices, however, requires adding further structure.

Another early representation theorem is now known as [[Stone's representation theorem for distributive lattices]] (the name honors [[Marshall Harvey Stone]], who first proved it). It characterizes distributive lattices as the lattices of [[Compact space|compact]] [[open set|open]] sets of certain [[topological space]]s. This result can be viewed both as a generalization of Stone's famous [[Stone's representation theorem for Boolean algebras|representation theorem for Boolean algebras]] and as a specialization of the general setting of [[Stone duality]].

A further important representation was established by [[Hilary Priestley]] in her [[Priestley's representation theorem for distributive lattice|representation theorem for distributive lattices]]. In this formulation, a distributive lattice is used to construct a topological space with an additional partial order on its points, yielding a (completely order-separated) ''ordered [[Stone's representation theorem for Boolean algebras|Stone space]]'' (or ''[[Priestley space]]''). The original lattice is recovered as the collection of [[clopen set|clopen]] lower sets of this space.

As a consequence of Stone's and Priestley's theorems, one easily sees that any distributive lattice is really isomorphic to a lattice of sets. However, the proofs of both statements require the [[Boolean prime ideal theorem]], a weak form of the [[axiom of choice]].

== Free distributive lattices ==
[[File:Monotone Boolean functions.svg|thumb|360px|Free distributive lattices on zero, one, two, and three generators. The elements labeled "0" and "1" are the empty join and meet, and the element labeled "majority" is (''x'' ∧ ''y'') ∨ (''x'' ∧ ''z'') ∨ (''y'' ∧ ''z'') = (''x'' ∨ ''y'') ∧ (''x'' ∨ ''z'') ∧ (''y'' ∨ ''z'').]]

The [[free object|free]] distributive lattice over a set of generators ''G'' can be constructed much more easily than a general free lattice. The first observation is that, using the laws of distributivity, every term formed by the binary operations <math>\lor</math> and <math>\land</math> on a set of generators can be transformed into the following equivalent ''normal form'':

:<math>M_1 \lor M_2 \lor \cdots \lor M_n,</math>

where <math>M_i</math> are finite meets of elements of ''G''. Moreover, since both meet and join are [[associative]], [[Commutativity|commutative]] and [[Idempotence|idempotent]], one can ignore duplicates and order, and represent a join of meets like the one above as a set of sets:

:<math>\{N_1, N_2, \ldots, N_n\},</math>

where the <math>N_i</math> are finite subsets of ''G''. However, it is still possible that two such terms denote the same element of the distributive lattice. This occurs when there are indices ''j'' and ''k'' such that <math>N_j</math> is a subset of <math>N_k.</math> In this case the meet of <math>N_k</math> will be below the meet of <math>N_j,</math> and hence one can safely remove the ''redundant'' set <math>N_k</math> without changing the interpretation of the whole term. Consequently, a set of finite subsets of ''G'' will be called ''irredundant'' whenever all of its elements <math>N_i</math> are mutually incomparable (with respect to the subset ordering); that is, when it forms an [[Sperner family|antichain of finite sets]].

Now the free distributive lattice over a set of generators ''G'' is defined on the set of all finite irredundant sets of finite subsets of ''G''. The join of two finite irredundant sets is obtained from their union by removing all redundant sets. Likewise the meet of two sets ''S'' and ''T'' is the irredundant version of <math>\{N \cup M \mid N \in S, M \in T\}.</math> The verification that this structure is a distributive lattice with the required [[universal property]] is routine.

The number of elements in free distributive lattices with ''n'' generators is given by the [[Dedekind number]]s. These numbers grow rapidly, and are known only for ''n''&nbsp;≤&nbsp;9; they are
:2, 3, 6, 20, 168, 7581, 7828354, 2414682040998, 56130437228687557907788, 286386577668298411128469151667598498812366 {{OEIS|id=A000372}}.
The numbers above count the number of elements in free distributive lattices in which the lattice operations are joins and meets of finite sets of elements, including the empty set. If empty joins and empty meets are disallowed, the resulting free distributive lattices have two fewer elements; their numbers of elements form the sequence
:0, 1, 4, 18, 166, 7579, 7828352, 2414682040996, 56130437228687557907786, 286386577668298411128469151667598498812364 {{OEIS|id=A007153}}.

==See also==
* [[Completely distributive lattice]] &mdash; a lattice in which infinite joins distribute over infinite meets
* [[Duality theory for distributive lattices]]
* [[Spectral space]]

==References==
{{reflist|2}}

==Further reading==

* {{cite book |last1=Burris |first1=Stanley N. |last2=Sankappanavar |first2=H.P. |title=A Course in Universal Algebra |url=http://www.thoralf.uwaterloo.ca/htdocs/ualg.html |year=1981 |publisher=Springer-Verlag |isbn=3-540-90578-2}}
* {{OEIS el|A006982|Number of unlabeled distributive lattices with ''n'' elements}}

{{Order theory}}

[[Category:Lattice theory]]