The Automorphism Group of the Finitary Power Monoid of the Integers under Addition
Salvatore Tringali, Kerou Wen
TL;DR
The paper addresses the problem of determining the automorphism group of the finitary power monoid $\\mathcal{P}_{\mathrm{fin}}(H)$, focusing on the additive integers. It develops a general framework using reduced quotients and minimizers for totally ordered abelian groups to translate automorphism questions to the non-negative cone, culminating in explicit automorphisms $f_\alpha$ and $g_\alpha$ and their negatives. For $H=\\mathbb{Z}_+$, it proves that every automorphism of $\\mathcal{P}_{\mathrm{fin}}(\\mathbb{Z})$ is of the form $\\pm f_\alpha$ or $\\pm g_\alpha$ with $\\alpha\\in\\mathbb{Z}$, and computes the automorphism group as ${\\rm Aut}(\\mathcal{P}_{\mathrm{fin}}(\\mathbb{Z}))\\cong \\mathbb{Z}_2 \\times \\mathrm{Dih}_\\infty$. This yields a precise symmetry classification for the finitary power monoid on the integers and informs related areas in factorization theory and automata; it also motivates an conjecture that similar inner-automorphism phenomena hold for reduced power monoids in more general totally ordered groups.
Abstract
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.