Definition:Total Ordering
Definition
Let $\RR \subseteq S \times S$ be a relation on a set $S$.
Definition 1
$\RR$ is a total ordering on $S$ if and only if:
That is, $\RR$ is an ordering with no non-comparable pairs:
- $\forall x, y \in S: x \mathop \RR y \lor y \mathop \RR x$
Definition 2
$\RR$ is a total ordering on $S$ if and only if:
| \(\text {(1)}: \quad\) | \(\ds \RR \circ \RR\) | \(\subseteq\) | \(\ds \RR\) | |||||||||||
| \(\text {(2)}: \quad\) | \(\ds \RR \cap \RR^{-1}\) | \(\subseteq\) | \(\ds \Delta_S\) | |||||||||||
| \(\text {(3)}: \quad\) | \(\ds \RR \cup \RR^{-1}\) | \(=\) | \(\ds S \times S\) |
Class Theory
In the context of class theory, the definition follows the same lines:
Let $V$ be a basic universe.
Let $\RR \subseteq V \times V$ be a relation.
Let $\RR$ be such that:
- $(1): \quad \RR$ is an ordering on $\Field \RR$
- $(2): \quad \forall x, y \in \Field \RR: x \mathop \RR y \lor y \mathop \RR x$ (that is, $x$ and $y$ are comparable)
where $\Field \RR$ denotes the field of $\RR$.
Then $\RR$ is a total ordering.
Also known as
Some sources refer to a total ordering as a linear ordering, or a simple ordering.
If it is necessary to emphasise that a total ordering $\preceq$ is not strict, then the term weak total ordering may be used.
Examples
Usual Ordering on Real Numbers
Let $\R$ denote the set of real numbers.
The usual ordering $\le$ on $\R$ and its dual $\ge$ are total orderings on $\R$.
Monarchy
Let $K$ denote the set of British monarchs.
Let $\MM$ denote the relation on $K$ defined as:
- $a \mathrel \MM b$ if and only if $a$ was monarch after or at the same time as $b$.
Its dual $\MM^{-1}$ is defined as:
- $a \mathrel {\MM^{-1} } b$ if and only if $a$ was monarch before or at the same time as $b$.
Then $\MM$ and $\MM^{-1}$ are total orderings on $K$.
Also see
- Results about total orderings can be found here.