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{fin}}(\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 $\mathcal P_{\mathrm{fin}}(\mathbb Z)$. In particular, we find that the automorphism group of $\mathcal P_{\mathrm{fin}}(\mathbb Z)$ is isomorphic to the direct product of a cyclic group of order two by the infinite dihedral group.