On automorphism group of the reduced finitary power monoid of the additive group of integers
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| Natura: | Preprint |
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2026
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| _version_ | 1866917288038367232 |
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| author | Wong, Dein Xu, Songnian Zhang, Chi Wang, Zhijun |
| author_facet | Wong, Dein Xu, Songnian Zhang, Chi Wang, Zhijun |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_19493 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On automorphism group of the reduced finitary power monoid of the additive group of integers Wong, Dein Xu, Songnian Zhang, Chi Wang, Zhijun Group Theory 08A35, 11P99, 20M13, 20M14 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. |
| title | On automorphism group of the reduced finitary power monoid of the additive group of integers |
| topic | Group Theory 08A35, 11P99, 20M13, 20M14 |
| url | https://arxiv.org/abs/2602.19493 |