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.