# Composite of Orderings is not necessarily Ordering

Jump to navigation
Jump to search

## 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 $14.27 \ \text {(c)}$