Posets, graphs and categories

Abstract

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