Partial Ordering/Examples/Ancestry
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Example of Partial Ordering
Let $P$ denote the set of all people who have ever lived.
Let $\DD$ denote the relation on $P$ defined as:
- $a \mathrel \DD b$ if and only if $a$ is a descendant of or the same person as $b$.
Its dual $\DD^{-1}$ is defined as:
- $a \mathrel {\DD^{-1} } b$ if and only if $a$ is an ancestor of or the same person as $b$.
Then $\DD$ and $\DD^{-1}$ are partial orderings on $P$.
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 14$: Orderings