Composite of Orderings is not necessarily Ordering
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Theorem
Let $A$ be a set.
Let $\RR$ and $\SS$ be orderings on $A$.
Then their composite $\RR \circ \SS$ is not necessarily also an ordering on $A$.
Proof
Let $\RR$ and $\SS$ be orderings as asserted.
Both $\RR$ and $\SS$ are a fortiori both antisymmetric and transitive.
But we have:
- From Composite of Antisymmetric Relations is not necessarily Antisymmetric, it is not necessarily the case that $\RR \circ \SS$ is itself antisymmetric.
- From Composite of Transitive Relations is not necessarily Transitive, it is not necessarily the case that $\RR \circ \SS$ is itself transitive.
Hence it is not necessarily the case that $\RR \circ \SS$ is an ordering.
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 14$: Orderings: Exercise $14.27 \ \text {(c)}$