Editing Distributive lattice (section)

== 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]].