Boundedly finite-to-one functions
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912594936201216 |
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| author | Hu, Xiao Shen, Guozhen |
| author_facet | Hu, Xiao Shen, Guozhen |
| contents | A function is boundedly finite-to-one if there is a natural number $k$ such that each point has at most $k$ inverse images. In this paper, we prove in $\mathsf{ZF}$ (i.e., the Zermelo--Fraenkel set theory without the axiom of choice) several results concerning this notion, among which are the following:
(1) For each infinite set $A$ and natural number $n$, there is no boundedly finite-to-one function from $\mathcal{S}(A)$ to $\mathcal{S}_{\leq n}(A)$, where $\mathcal{S}(A)$ is the set of all permutations of $A$ and $\mathcal{S}_{\leq n}(A)$ is the set of all permutations of $A$ moving at most $n$ points.
(2) For each infinite set $A$, there is no boundedly finite-to-one function from $\mathcal{B}(A)$ to $\mathrm{fin}(A)$, where $\mathcal{B}(A)$ is the set of all partitions of $A$ such that every block is finite and $\mathrm{fin}(A)$ is the set of all finite subsets of $A$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_10183 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Boundedly finite-to-one functions Hu, Xiao Shen, Guozhen Logic Combinatorics Primary 03E10, Secondary 03E25 A function is boundedly finite-to-one if there is a natural number $k$ such that each point has at most $k$ inverse images. In this paper, we prove in $\mathsf{ZF}$ (i.e., the Zermelo--Fraenkel set theory without the axiom of choice) several results concerning this notion, among which are the following: (1) For each infinite set $A$ and natural number $n$, there is no boundedly finite-to-one function from $\mathcal{S}(A)$ to $\mathcal{S}_{\leq n}(A)$, where $\mathcal{S}(A)$ is the set of all permutations of $A$ and $\mathcal{S}_{\leq n}(A)$ is the set of all permutations of $A$ moving at most $n$ points. (2) For each infinite set $A$, there is no boundedly finite-to-one function from $\mathcal{B}(A)$ to $\mathrm{fin}(A)$, where $\mathcal{B}(A)$ is the set of all partitions of $A$ such that every block is finite and $\mathrm{fin}(A)$ is the set of all finite subsets of $A$. |
| title | Boundedly finite-to-one functions |
| topic | Logic Combinatorics Primary 03E10, Secondary 03E25 |
| url | https://arxiv.org/abs/2407.10183 |