Totally ordered set

A totally ordered set is a pair $(S,\le )$ of a set $S$ and an order relation $\le$ on $S$, which 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.
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.

The order relation in a totally ordered set is called a total order (hence the name), which can be distinguished from a partially ordered set in that the latter only has a partial order which does not satisfy the third property, but a weaker version where $a\le a$ for all $a$.

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.