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distributive lattice

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Context

(0,1)-Category theory

(0,1)-category theory: logic, order theory

(0,1)-category

(0,1)-topos

Theorems

Contents

Definition

Definition

A distributive lattice is a lattice in which join ∨ and meet ∧ distribute over each other, in that for all x,y,z in the latiice, the distributivity laws are satisfied:

  • x∨(y∧z)=(x∨y)∧(x∨z),
  • x∧(y∨z)=(x∧y)∨(x∧z).
Remark

The nullary forms of distributivity follow automatically:

  • x∨⊤=⊤,
  • x∧⊥=⊥.

Distributive lattices and lattice homomorphisms form a concrete category DistLat.

Remark

Any lattice that satisfies one of the two binary distributivity laws must also satisfy the other; isn't that nice? This convenience does not extend to infintary distributivity, however.

Examples

Any Boolean algebra, and even any Heyting algebra, is a distributive lattice.

Properties

Categorification

Every distributive lattice, regarded as a category (a (0,1)-category), is a coherent category.

Conversely, the notion of coherent category may be understood as a categorification of the notion of distributive lattices.

Completion

The completely distributive algebraic lattices (the frames of opens of Alexandroff locales ) form a reflective subcategory of that of all distributive lattices. The reflector is called canonical extension.

Revised on March 15, 2012 17:58:26 by Urs Schreiber (82.169.65.155)
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