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.

Any linear order is a distributive lattice.

An integral domain is a Prüfer domain? iff its lattice of ideals is distributive.

Properties

Finite distributive lattices

Let FinDistLatFinDistLat be the category of finite distributive lattices and lattice homomorphisms, and let FinPosetFinPoset be the category of finite posets and order-preserving functions. These are contravariantly equivalent, thanks to the presence of an ambimorphic object:

Proposition. The opposite category of FinDistLatFinDistLat is equivalent to FinPosetFinPoset:

FinDistLat opFinPoset. FinDistLat^{op} \simeq FinPoset \,.

This equivalence is given by the functor

[,2]:FinDistLat opFinPoset [-,2] \;\colon\; FinDistLat^{op} \stackrel{\simeq}{\to} FinPoset

where 22 is the 2-element distributive lattice, and

[,2]:FinPoset opFinDistLat [-,2] \;\colon\; FinPoset^{op} \stackrel{\simeq}{\to} FinDistLat

where 2={0,1}2 = \{0,1\} is the 2-element poset with 0<10 \lt 1.

This is mentioned in

  • Gavin C. Wraith, Using the generic interval, Cah. Top. Géom. Diff. Cat. XXXIV 4 (1993) pp.259-266. (pdf)

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 June 6, 2015 20:54:19 by John Baez (99.11.156.244)