# Totally ordered set

https://arbital.com/p/totally_ordered_set

by Joe Zeng Jul 5 2016 updated Jul 21 2016

A set where all the elements can be compared as greater than or less than.

[summary: A totally ordered set is a pair $(S, \le)$ of a set $S$ and a total order $\le$ on $S$, which is a binary relation that satisfies the following properties:

1. For all $a, b \in S$, if $a \le b$ and $b \le a$, then $a = b$. (the antisymmetric property)
2. For all $a, b, c \in S$, if $a \le b$ and $b \le c$, then $a \le c$. (the transitive property)
3. For all $a, b \in S$, either $a \le b$ or $b \le a$, or both. (the [total_relation totality] property)]

A totally ordered set is a pair $(S, \le)$ of a set $S$ and a total order $\le$ on $S$, which is a [-binary_relation] that satisfies the following properties:

1. For all $a, b \in S$, if $a \le b$ and $b \le a$, then $a = b$. (the antisymmetric property)
2. For all $a, b, c \in S$, if $a \le b$ and $b \le c$, then $a \le c$. (the transitive property)
3. For all $a, b \in S$, either $a \le b$ or $b \le a$, or both. (the [total_relation totality] 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

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.

A totally ordered set $(S, \\le)$ is a set $S$ with a comparison operator $\\le$ that is defined for all members of $S$\.
Correct me if I'm wrong, but isn't it idiosyncratic to define $\leq$ as a predicate rather than a relation? I know of at least three books that describe it as a relation: The Joy of Sets by Devlin, Principles of Mathematical Analysis by Rudin, and Introduction to Lattice and Order by Davey and Priestly.
Also, isn't $\leq$ called an order rather than a comparison?