Total Ordering/Examples

Examples of Total Orderings

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 $K$.