A totally ordered set is a pair of a set and a total order on , which is a binary_relation that satisfies the following properties:
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 for all .
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.
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.