Consider a setvalued, finitevalued map $F$ from a set $X$ to subsets of $X$. Consider the following property: $F(x) \geq F(y)$ for all $x,y$ such that $y \in F(x)$. I have defined this property myself in a specific context but I am not sure what name to give it. I would like to know if there is a standard name for this property or similar property in setvalued analysis or otherwise. Any suggestions for names would also be appreciated.
