so I have a binary function $f: P \times P \to \mathbb{R}$ over some power set $P$. I also have a partial ordering $(P,≤)$ over $P$ (by set size). For this function I know that
$f(a,b) ≥ 0 \implies f(a',b) ≥ 0 \quad \forall a,a',b \in P \quad \text{and} \quad a ≤ a'$
What other conditions does $f$ need to satisfy in order for it to be a monotone over the partial order in the first argument?
Cheers, Paul