On power monoids and their automorphisms

Endowed with the binary operation of set addition, the family Pfin,0(N) of all finite subsets of N containing 0 forms a monoid, with the singleton {0} as its neutral element. We show that the only non-trivial automorphism of Pfin,0(N) is the involution X↦max⁡X−X. The proof leverages ideas from additive number theory and proceeds through an unconventional induction on what we call the boxing dimension of a finite set of integers, that is, the smallest number of (discrete) intervals whose union is the set itself.