Hasse Diagram/Examples/British Monarchs

Example of Hasse Diagram

Recall the ordering on the British 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 $K$.

This Hasse diagram illustrates the restriction of $\MM$ to all $x$ of $K$ such that $x \mathrel \MM \text {Victoria}$ and $\text {Elizabeth II} \mathrel \MM x$.

The corresponding Hasse diagram illustrating the restriction of $\MM^{-1}$ to the same set is given below:

Source of Name

This entry was named for Helmut Hasse.

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 14$: Orderings: Figure $12 \ (1)$
• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 14$: Orderings: Exercise $14.1$