(0,1)-category

(0,1)-topos

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Definition

Definition

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

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

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.

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)