Boundedly finite-to-one functions

TL;DR

This paper investigates properties of boundedly finite-to-one functions within set theory without the axiom of choice, proving their non-existence between certain infinite sets related to permutations and partitions.

Contribution

It establishes new results on the non-existence of boundedly finite-to-one functions between specific infinite set classes in ZF set theory.

Findings

No boundedly finite-to-one function from permutations to limited-move permutations.

No boundedly finite-to-one function from partitions to finite subsets.

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 $\mathsf{ZF}$ (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 $A$.