# Definition:Distributive Lattice

## Definition

Let $\left({S, \vee, \wedge, \preceq}\right)$ be a lattice.

Then $\left({S, \vee, \wedge, \preceq}\right)$ is distributive if and only if one (hence all) of the following equivalent statements holds:

 $(1)$ $:$ $\displaystyle \forall x, y, z \in S:$ $\displaystyle x \wedge \left({y \vee z}\right) = \left({x \wedge y}\right) \vee \left({x \wedge z}\right)$ $(1')$ $:$ $\displaystyle \forall x, y, z \in S:$ $\displaystyle \left({x \vee y}\right) \wedge z = \left({x \wedge z}\right) \vee \left({y \wedge z}\right)$ $(2)$ $:$ $\displaystyle \forall x, y, z \in S:$ $\displaystyle x \vee \left({y \wedge z}\right) = \left({x \vee y}\right) \wedge \left({x \vee z}\right)$ $(2')$ $:$ $\displaystyle \forall x, y, z \in S:$ $\displaystyle \left({x \wedge y}\right) \vee z = \left({x \vee z}\right) \wedge \left({y \vee z}\right)$

That is, $\left({S, \vee, \wedge, \preceq}\right)$ is distributive if and only if $\wedge$ and $\vee$ distribute over each other.

## Also see

• Results about distributive lattices can be found here.