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Suppose I have a function $f$ on sets.

What is the property of $f$ called when,

for all sets $x$, $y$: $f(x)$ is a superset of $f(y)$ when $x$ is a superset of $y$

i.e.

$$\forall x,y : x\supseteq y \Rightarrow f(x) \supseteq f(y)$$

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    $\begingroup$ Is this a question of computer science? $\endgroup$ – J.-E. Pin Mar 24 '15 at 10:58
  • $\begingroup$ I think it's not; this seems to be pure mathematics question which we should probably migrate over to Mathematics. Community votes, please! (cc @J.-E.Pin) $\endgroup$ – Raphael Mar 24 '15 at 14:42
  • $\begingroup$ Come on, guys, monotonicity is super important in lattice theory which has vast applications in logic, coalgebra, in the study of submodular functions and therefore also matroids, and also in the field of circuit complexity. What's with the trigger happiness? $\endgroup$ – Pål GD Mar 25 '15 at 12:48
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It is called monotonicity with respect to the inclusion ordering of sets.

More precisely, it is in this case increasing monotonicity since the order is preserved. If the order was reversed, it would be decreasing monotonicity.

It can also be called direct monotonicity and reverse monotonicity, as increasing/decreasing seems used mostly for numeric functions.

The function is said to be strictly monotone iff (in the direct case)

$$\forall x,y : x\supsetneq y \Rightarrow f(x) \supsetneq f(y)$$

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    $\begingroup$ What you're calling "decreasing monotonicity" (a term I've not heard) is also called being anti-monotone or antitone. $\endgroup$ – David Richerby Mar 24 '15 at 12:58
  • $\begingroup$ @DavidRicherby "decreasing monotonicity" is used mostly for numeric functions. But I did not see why it should be restricted to that. I prefer direct and reverse monotonicity. I am not too happy with anti-monotone as it can be inperpreted as the opposite of monotone, while reverse nonotonicity is still monotonicity. I never heard of antitone, but why not. But I will not fight for terminology, unless it has political/sociological impact. $\endgroup$ – babou Mar 24 '15 at 13:06

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