Category:Distributive Lattices
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This category contains results about Distributive Lattices.
Definitions specific to this category can be found in Definitions/Distributive Lattices.
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:
\((1)\) | $:$ | \(\ds \forall x, y, z \in S:\) | \(\ds x \wedge \paren {y \vee z} = \paren {x \wedge y} \vee \paren {x \wedge z} \) | ||||||
\((1')\) | $:$ | \(\ds \forall x, y, z \in S:\) | \(\ds \paren {x \vee y} \wedge z = \paren {x \wedge z} \vee \paren {y \wedge z} \) | ||||||
\((2)\) | $:$ | \(\ds \forall x, y, z \in S:\) | \(\ds x \vee \paren {y \wedge z} = \paren {x \vee y} \wedge \paren {x \vee z} \) | ||||||
\((2')\) | $:$ | \(\ds \forall x, y, z \in S:\) | \(\ds \paren {x \wedge y} \vee z = \paren {x \vee z} \wedge \paren {y \vee z} \) |
Subcategories
This category has the following 2 subcategories, out of 2 total.
B
- Boolean Lattices (6 P)
W
- Well Inside Relation (6 P)
Pages in category "Distributive Lattices"
The following 10 pages are in this category, out of 10 total.