Definition:Proper Subtower in Set

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Let $X$ be a set.

Let $\left({T, \preccurlyeq}\right)$ be a tower in $X$.

Then $\left({T, \preccurlyeq}\right)$ is a proper subtower in $X$ if and only if:

$T$ is a proper subset of some set $T' \subseteq X$


$\left({T', \preccurlyeq}\right)$ is a tower in $X$.

Historical Note

The name proper subtower in set has been specifically coined for $\mathsf{Pr} \infty \mathsf{fWiki}$, and may or may not appear in the literature.