The Automorphism Group of the Finitary Power Monoid of the Integers under Addition

Endowed with the binary operation of set addition carried over from the integers, the family ${\mathcal{P}}_{\mathrm{f}\mathrm{i}\mathrm{n}}(\mathbb{Z})$ of all non-empty finite subsets of $\mathbb{Z}$ forms a monoid whose neutral element is the singleton $\{0\}$. Building upon recent work by Tringali and Yan, we determine the automorphisms of ${\mathcal{P}}_{\mathrm{f}\mathrm{i}\mathrm{n}}(\mathbb{Z})$. In particular, we find that the automorphism group of ${\mathcal{P}}_{\mathrm{f}\mathrm{i}\mathrm{n}}(\mathbb{Z})$ is isomorphic to the direct product of a cyclic group of order two by the infinite dihedral group.