- Reflexivity: x is less than or equal to x;
- Antisymmetry: If x is less than or equal to y and y is less than or equal to x, then x equals y; and
- Transitivity: If x is less than or equal to y and y is less than or equal to z, then x is less than or equal to z.

A chain in P is a subset of P where every two members of the subset are comparable.

An antichain in P is a subset of P where no two members of the subset are comparable (unless they are equal).

The width of a finite partial order is the size of its largest antichain.

Dilworth's theorem says that the width of P is equal to the smallest n such that P can be written as the union of n chains.

I keep nibbling away at the proof of this one, but it's slippery.

**Offering:**

Doors are solemn things, guarding entrances,

Opening to the favored,

Closing to thieves and weather.

Doors are solemn things, but

Cat doors are for cats.