On automorphism group of the reduced finitary power monoid of the additive group of integers

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 $X \mapsto \beta(X) - X$, 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.