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.
Categories: math.GR