Maximum of element weights ($f(S)={\max }_{v\in S}w(v)$)

On nonempty subsets of a nonempty finite set $V$, take the maximum of the contained element weights. Setting $f(\varnothing )={\min }_{v\in V}w(v)$ extends this to a monotone set function and a submodular set function. More generally any value at the empty set at most every element weight works, because ${\Delta }_{x}f(S)=\max \{0,w(x)-f(S)\}$ satisfies diminishing returns.