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How would you name a following property of the function $T$?

$$ \forall a,b\;(a \subset b \implies T(a) \subset T(b)) $$

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    $\begingroup$ Welcome to Mathematics Stack Exchange. $a$ and $b$ are subsets of $T$'s domain? $\endgroup$ – J. W. Tanner Oct 25 at 20:42
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In axiomatic set theory, these functions are usually called (monotone) increasing.

In order theory, the term monotone is used for the property $a\prec b \Rightarrow f(a) \prec f(b)$, whatever the $\prec$ relation is (be it $\leqslant$ or $\subseteq$, or some other order relation), but I'd be careful with the terminology since in most cases (especially in things like analysis, calculus, etc.) the terms monotone, increasing and decreasing usually refer to this property with the usual relation $\leqslant$ on $\mathbb R$.

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I would call that order-preserving under the subset order.

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Monotone, ordering by inclusion.

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