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Consider a set-valued, finite-valued 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 set-valued analysis or otherwise. Any suggestions for names would also be appreciated.

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Isn't this is just some kind of monotonicity? –  Suvrit Aug 22 '12 at 5:59
    
It is monotonicity along certain paths in the graph of F. –  Ankur Aug 22 '12 at 14:18

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