I went through a theorem which is given below- "A function S(X) operating on a set X can be a valid scoring function, i.e. it is permutation invariant to the elements in X, if and only if it can be decomposed in the form ρ(∑x∈X φ(x)) , for suitable transformations φ and ρ." The theorem can be found here - https://arxiv.org/abs/1703.06114
Since Max(X) is obviously a permutation invariant function i.e it is not affected by order of elements in a set hence it can be broken down in this form. So what will be φ and ρ in this case?