Editing Distributive lattice (section)

== 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"/>