Totally ordered set

A totally ordered set $(S,\le )$ is a set $S$ with a comparison operator $\le$ that is defined for all members of $S$.

A comparison operator on any set $S$ is a binary predicate $S^2\to \{\text{true},\text{false}\}$ that satisfies the following properties:

1. For all $a,b\in S$, if $a\le b$ and $b\le a$, then $a=b$. This is known as the antisymmetric property.
2. For all $a,b,c\in S$, if $a\le b$ and $b\le c$, then $a\le c$. This is known as the transitive property.

A comparison operator for a totally ordered set (also known as a total_order on that set, hence the name) satisfies one more property:

3. For all $a,b\in S$, either $a\le b$ or $b\le a$, or both. This is the totality property, which is equivalent to the trichotomy property.

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.