On power monoids and their automorphisms

Abstract

Endowed with the binary operation of set addition, the family ${\mathcal{P}}_{\mathrm{fin},0}(\mathbb{N})$ 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 ${\mathcal{P}}_{\mathrm{fin},0}(\mathbb{N})$ 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.

11 pages, no figures. Fixed many typos and a couple of mistakes from the previous version. To appear in J Combinatorial Theory Ser A