nLab
distributive lattice

Contents

Definition

Definition

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

  • x(yz)=(xy)(xz)x \vee (y \wedge z) = (x \vee y) \wedge (x \vee z),
  • x(yz)=(xy)(xz)x \wedge (y \vee z) = (x \wedge y) \vee (x \wedge z).
Remark

The nullary forms of distributivity follow automatically:

  • x=x \vee \top = \top,
  • x=x \wedge \bot = \bot.

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 infinitary 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 May 26, 2013 17:04:04 by Christoph Rauch? (188.195.32.157)