Ordering/Examples

Examples of Orderings

Integer Difference on Reals

Let $\preccurlyeq$ denote the relation on the set of real numbers $\R$ defined as:

$a \preccurlyeq b$ if and only if $b - a$ is a non-negative integer

Then $\preccurlyeq$ is an ordering on $\R$.

Example Ordering on Integers

Let $\preccurlyeq$ denote the relation on the set of integers $\Z$ defined as:

$a \preccurlyeq b$ if and only if $0 \le a \le b \text { or } b \le a < 0 \text { or } a < 0 \le b$

where $\le$ is the usual ordering on $\Z$.

Then $\preccurlyeq$ is an ordering on $\Z$.

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 orderings on $K$.

American Presidency

Let $S$ denote the set of American presidents.

Let $\PP$ denote the relation on $S$ defined as:

$a \mathrel \PP b$ if and only if $a$ was president after or at the same time as $b$.

Because Grover Cleveland was president both before and after Benjamin Harrison, $\PP$ is not an antisymmetric relation.

Thus $\PP$ is not an ordering on $S$.