On automorphism group of the reduced finitary power monoid of the additive group of integers
Dein Wong, Songnian Xu, Chi Zhang, Zhijun Wang
TL;DR
This work resolves the automorphism structure of the reduced finitary power monoid $P_{fin,0}(\mathbb{Z})$ under setwise addition by proving that the only nontrivial automorphism is the negation map $X\mapsto -X$; it achieves this by analyzing how automorphisms act on small base sets, introducing a reversal operation and performing a min/max–based, boxing-dimension–driven induction to eliminate all nonidentity possibilities. The proof hinges on reducing to the case where $\{0,1\}$ is fixed and then showing any such automorphism must fix all of $P_{fin,0}(\mathbb{Z})$ or coincide with the negation, thereby completing the conjecture in the literature. Consequently, the automorphism group of the reduced finitary power monoid of $(\mathbb{Z},+)$ is of order two, generated by the identity and the negation map. This advances understanding of automorphisms in reduced power semigroups and informs applications in additive combinatorics and related algebraic structures.
Abstract
Let $\mathbb{Z}$ be the additive group of all integers and $\mathbb{N}$ the sub-monoid of $\mathbb{Z}$ of all non-negative integers. For a finite subset $X$ of $\mathbb{Z}$, we denote by ${\rm max}\ X$ the maximum member in $X$. %Recently, Tringali and Yan (\cite{tri2}, J. Combin. Theory Ser. A, 209(2025)) proved that the only non-trivial automorphism of $\mathcal{P}_{{\rm fin,} 0}(\mathbb{N})$ %is the involution $X \mapsto β(X) - X$, and they posed a conjecture: {\it The automorphism group of the reduced power monoid $\mathcal{P}_{{\rm fin,} 0}(S)$ of a numerical %monoid $S$ properly contained in $\mathbb{N}$ must be the identity}. Recently, Tringali and Yan (\cite{tri2}, J. Comb. Theory, Ser. A, 209(2025)) proved that the only non-trivial automorphism of $\mathcal{P}_{{\rm fin,} 0}(\mathbb{N})$ is the involution $X \mapsto {\rm max}\ X - X$. Following up on the result in \cite{tri2}, Tringali and Wen \cite{triwen} proved that the automorphism group of the power monoid $\mathcal{P}_{\rm fin}(\mathbb{Z})$ is isomorphic to $\mathbb{Z}_2 \times {\rm Dih}_{\infty}$, where ${\rm Dih}_{\infty}$ refers to the infinite dihedral group. At the end part of \cite{triwen}, Tringali and Wen left a conjecture as follows: {\it The only non-trivial automorphism of the reduced finitary power monoid of $(\mathbb{Z},+)$ is given by $X\mapsto -X$.} In the present paper, we aim to give a positive proof for the above conjecture.