# What is the name of the property where $f(A) \supseteq f(B)$ when $A\supseteq B$?

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.

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

• Is this a question of computer science? – J.-E. Pin Mar 24 '15 at 10:58
• 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) – Raphael Mar 24 '15 at 14:42
• 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? – Pål GD Mar 25 '15 at 12:48

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