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 that satisfies the following properties:
For all \(a, b \in S\), if \(a \le b\) and \(b \le a\), then \(a = b\). (the property)
For all \(a, b, c \in S\), if \(a \le b\) and \(b \le c\), then \(a \le c\). (the transitive property)
For all \(a, b \in S\), either \(a \le b\) or \(b \le a\), or both. (the 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
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.