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Xiao Hu, Guozhen Shen, Boundedly finite-to-one functions, Logic Journal of the IGPL, 2025;, jzae130, https://doi.org/10.1093/jigpal/jzae130
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Abstract
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 |$\textsf{ZF}$| (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 |$\textrm{fin}(A)$|, where |$\mathcal{B}(A)$| is the set of all partitions of |$A$| such that every block is finite and |$\textrm{fin}(A)$| is the set of all finite subsets of |$A$|.
