Axiom:Distributive Lattice Axioms
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Definition
Let $\struct {S, \vee, \wedge, \preceq}$ be a lattice.
$\struct {S, \vee, \wedge, \preceq}$ is a distributive lattice if and only if $\struct {S, \vee, \wedge, \preceq}$ satisfies the axioms:
\((1)\) | $:$ | \(\ds \forall x, y, z \in S:\) | \(\ds x \wedge \paren {y \vee z} = \paren {x \wedge y} \vee \paren {x \wedge z} \) | ||||||
\((1')\) | $:$ | \(\ds \forall x, y, z \in S:\) | \(\ds \paren {x \vee y} \wedge z = \paren {x \wedge z} \vee \paren {y \wedge z} \) | ||||||
\((2)\) | $:$ | \(\ds \forall x, y, z \in S:\) | \(\ds x \vee \paren {y \wedge z} = \paren {x \vee y} \wedge \paren {x \vee z} \) | ||||||
\((2')\) | $:$ | \(\ds \forall x, y, z \in S:\) | \(\ds \paren {x \wedge y} \vee z = \paren {x \vee z} \wedge \paren {y \vee z} \) |
These criteria are called the distributive lattice axioms.