Introduction to order

The idea of an order is central to many kinds of mathematics. The real numbers are familiarly ordered as a number–line, and even a collection of sets will be seen to be partially ordered by the ‘subset of’ relation. We shall start by presenting the axioms for a partially ordered set and then discuss one particularly interesting question about such sets, whether they have maximal elements.

An order relation is a relation R between elements x, y of some set X, where xRy means x is smaller than or comes before y. An alternative notation arises when one thinks of the relation more concretely as a set of pairs (x, y), a subset of X2 = {(x, y) : x, y ∈ X}. We can then write xRy in an alternative way as (x, y) ∈ R.

Definition 2.1 A partial order on a set X is a relation R ⊆ X2 such that

1. (i) (x, y) ∈ R and (y, z) ∈ R implies (x, z) ∈ R

2. (ii) (x, x) ∉ R

for all x, y ∈ X.

Example 2.2 The relation on the set of real numbers ℝ defined by ‘(x, y) ∈ R if and only if x < y’ is a partial order, where < is the usual order on the set of real numbers. In fact it is a special kind of partial order that we will later call a total order or linear order.