Studying the divisibility of power LCM matrics by power GCD matrices on gcd-closed sets
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866910771476168704 |
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| author | Zhao, Jianrong Wang, Chenxu Fu, Yu |
| author_facet | Zhao, Jianrong Wang, Chenxu Fu, Yu |
| contents | Let $S=\{x_1,\ldots, x_n\}$ be a gcd-closed set (i.e. $(x_i,x_j)\in S $ for all $1\le i,j\le n$). In 2002, Hong proposed the divisibility problem of characterizing all gcd-closed sets $S$ with $|S|\ge 4$ such that the GCD matrix $(S)$ divides the LCM matrix $[S]$ in the ring $M_{n}(\mathbb{Z})$. For $x\in S,$ let $G_S(x):=\{z\in S: z<x, z|x \text{ and } (z|y|x, y\in S)\Rightarrow y\in\{z,x\}\}$. In 2009, Feng, Hong and Zhao answered this problem in the context where $\max_{x \in S}\{|G_S(x)|\} \leq 2$. In 2022, Zhao, Chen and Hong obtained a necessary and sufficient condition on the gcd-closed set $S$ with $\max_{x \in S}\{|G_S(x)|\}=3$ such that $(S)|\left[S\right].$ Meanwhile, they raised a conjecture on the necessary and sufficient condition such that $(S)|\left[S\right]$ holds for the remaining case $\max_{x \in S}\{|G_S(x)|\}\ge 4$. In this papar, we confirm the Zhao-Chen-Hong conjecture from a novel perspective, consequently solve Hong's open problem completely. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_01794 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Studying the divisibility of power LCM matrics by power GCD matrices on gcd-closed sets Zhao, Jianrong Wang, Chenxu Fu, Yu Combinatorics Let $S=\{x_1,\ldots, x_n\}$ be a gcd-closed set (i.e. $(x_i,x_j)\in S $ for all $1\le i,j\le n$). In 2002, Hong proposed the divisibility problem of characterizing all gcd-closed sets $S$ with $|S|\ge 4$ such that the GCD matrix $(S)$ divides the LCM matrix $[S]$ in the ring $M_{n}(\mathbb{Z})$. For $x\in S,$ let $G_S(x):=\{z\in S: z<x, z|x \text{ and } (z|y|x, y\in S)\Rightarrow y\in\{z,x\}\}$. In 2009, Feng, Hong and Zhao answered this problem in the context where $\max_{x \in S}\{|G_S(x)|\} \leq 2$. In 2022, Zhao, Chen and Hong obtained a necessary and sufficient condition on the gcd-closed set $S$ with $\max_{x \in S}\{|G_S(x)|\}=3$ such that $(S)|\left[S\right].$ Meanwhile, they raised a conjecture on the necessary and sufficient condition such that $(S)|\left[S\right]$ holds for the remaining case $\max_{x \in S}\{|G_S(x)|\}\ge 4$. In this papar, we confirm the Zhao-Chen-Hong conjecture from a novel perspective, consequently solve Hong's open problem completely. |
| title | Studying the divisibility of power LCM matrics by power GCD matrices on gcd-closed sets |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2501.01794 |