On power monoids and their automorphisms

Abstract

Endowed with the binary operation of set addition, the family $P_{\mathrm{fin},0}\left(N\right)$ 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 $P_{\mathrm{fin},0}\left(N\right)$ 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.