distributive lattice

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

- $x \vee (y \wedge z) = (x \vee y) \wedge (x \vee z)$,
- $x \wedge (y \vee z) = (x \wedge y) \vee (x \wedge z)$.

The nullary forms of distributivity follow automatically:

- $x \vee \top = \top$,
- $x \wedge \bot = \bot$.

Distributive lattices and lattice homomorphisms form a concrete category DistLat.

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.

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.

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.

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 10, 2015 15:05:40
by Todd Trimble
(67.81.95.215)