Definition:Distributive Lattice

From ProofWiki
Jump to navigation Jump to search

Definition

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


Definition 1

Then $\struct {S, \vee, \wedge, \preceq}$ is distributive if and only if $\struct {S, \vee, \wedge, \preceq}$ satisfies one of the distributive lattice axioms:

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


Definition 2

Then $\struct {S, \vee, \wedge, \preceq}$ is distributive if and only if $\struct {S, \vee, \wedge, \preceq}$ satisfies all of the distributive lattice axioms:

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


Also see

  • Results about distributive lattices can be found here.