Definition:Distributive Lattice

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Definition

Let $\struct {S, \vee, \wedge, \preceq}$ be a lattice.


Then $\struct {S, \vee, \wedge, \preceq}$ is distributive if and only if one of the following statements holds:

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


That these statements are in fact equivalent is shown on Equivalence of Definitions of Distributive Lattice.

Hence, $\struct {S, \vee, \wedge, \preceq}$ is distributive if and only if $\wedge$ and $\vee$ distribute over each other.


Also see

  • Results about distributive lattices can be found here.