# Definition:Distributive Lattice

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## 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.