On power monoids and their automorphisms

Abstract

Endowed with the binary operation of set addition, the family $\mathcal P_{{\rm fin},0}(\mathbb N)$ of all finite subsets of $\mathbb N$ containing $0$ forms a monoid, with the singleton $\{0\}$ as its neutral element. We show that the only non-trivial automorphism of $\mathcal P_{{\rm fin},0}(\mathbb N)$ is the involution $X \mapsto \max X - X$. 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.