Editing Distributive lattice (section)

==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)]].