In Appendix A, we recall some of the basic concepts associated with partially ordered sets, graphs and categories.

Posets and graphs

A partial order on a set P is a binary relation ≤ satisfying the following three properties:

1. (i) reflexivity: for all x ∈ P, we have x ≤ x;

2. (ii) antisymmetry: for all x, y ∈ P, if we have both x ≤ y and y ≤ x, then x = y;

3. (iii) transitivity: for all x, y, z ∈ P, if we have both x ≤ y and y ≤ z, then x ≤ z.

A set P equipped with a partial order ≤ is known as a partially ordered set or poset. If Q is a subset of a poset P, then Q inherits a poset structure from P by restricting the relation ≤ to Q.

Strictly speaking, a partial order is the subset of P × P given by

{(x, y) ∈ P × P : x ≤ y}.

Every partial order, ≤, on P has an opposite order on P, denoted by ≥ This is the subset of P × P with the property that (y, x) ∈ ≤ if and only if (x, y) ∈ ≥ It turns out (Exercise A.1.4) that ≤ is also a partial order. We write P. to refer to the set P equipped with the opposite partial order.