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

TL;DR

This paper proves that the only non-trivial automorphism of the reduced finitary power monoid of the integers is the negation map, confirming a conjecture posed by previous researchers.

Contribution

It provides a positive proof that the automorphism group of the reduced finitary power monoid of integers is generated solely by the negation map.

Findings

The automorphism group is generated by the negation map.

The only non-trivial automorphism is the map $X o -X$.

Confirms the conjecture by Tringali and Wen.

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.