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