Total Ordering/Examples
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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 $K$.