Totally ordered set

A totally ordered set is a pair \((S, \le)\) of a set \(S\) and a total order \(\le\) on \(S\), which is a binary relation that satisfies the following properties:

1. For all \(a, b \in S\), if \(a \le b\) and \(b \le a\), then \(a = b\). (the antisymmetric property)

2. For all \(a, b, c \in S\), if \(a \le b\) and \(b \le c\), then \(a \le c\). (the transitive property)

3. For all \(a, b \in S\), either \(a \le b\) or \(b \le a\), or both. (the totality property)

A totally ordered set is a special type of partially ordered set that satisfies the total property — in general, posets only satisfy the reflexive property, which is that \(a \le a\) for all \(a \in S\).

Examples of totally ordered sets

The real numbers are a totally ordered set. So are any of the subsets of the real numbers, such as the rational numbers or the integers.

Examples of not totally ordered sets

The complex numbers do not have a canonical total ordering, and especially not a total ordering that preserves all the properties of the ordering of the real numbers, although one can define a total ordering on them quite easily.