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 lattice, 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.

Let $FinDistLat$ be the category of finite distributive lattices and lattice homomorphisms, and let $FinPoset$ 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 $FinDistLat$ is equivalent to $FinPoset$:

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

This equivalence is given by the functor

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

where $2$ is the 2-element distributive lattice, and

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

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

This is mentioned in

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

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 lattice. A different categorification is the notion of distributive category.

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 3, 2016 15:27:39
by Mike Shulman
(192.195.154.58)