Definition:Inclusion-Reversing Mapping

Definition
Let $A, B$ be sets of sets and $\phi: A \to B$ be a mapping.

Then $\phi$ is inclusion-reversing :
 * for every pair of sets $a_1, a_2 \in A$ such that $a_1 \subseteq a_2$:
 * $\phi \left({a_2}\right) \subseteq \phi \left({a_1}\right)$

Also see

 * Definition:Inclusion-Preserving Mapping

Generalizations

 * When $\left({A, \subseteq}\right)$ and $\left({B, \subseteq}\right)$ are ordered by inclusion, an inclusion-reversing mapping can be considered as a decreasing mapping.
 * Definition:Contravariant Functor