Editing Distributivity (order theory) (section)

==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]]).