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:

1. For all $a,b\in S$, if $a\le b$ and $b\le a$, then $a=b$. (the property)
2. For all $a,b,c\in S$, if $a\le b$ and $b\le c$, then $a\le c$. (the property)
3. 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 that satisfies the total property — in general, posets only satisfy the property, which is that $a\le a$ for all $a\in S$.

Examples of totally ordered sets

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

Examples of not totally ordered sets

The 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.