# Decompositions of set-valued mappings

## TL;DR

This paper proves that certain set-valued mappings with bounded fibers can be decomposed into a small family of bijective selectors, with applications to the structure of balleans.

## Contribution

It introduces a decomposition theorem for set-valued mappings with bounded fibers, providing a new way to represent such mappings via a small family of bijective selectors.

## Key findings

- Existence of a small family of bijective selectors for set-valued mappings.
- Decomposition of set-valued mappings into single-valued functions.
- Application to G-space representations of balleans.

## Abstract

Let $X$ be a set, $B_{X}$ denotes the family of all subsets of $X$ and $F: X \longrightarrow B_{X}$ be a set-valued mapping such that $x \in F(x)$, $sup_{x\in X} | F(x)|< \kappa$, $sup_{x\in X} | F^{-1}(x)|< \kappa$ for all $x\in X$ and some infinite cardinal $\kappa$. Then there exists a family $\mathcal{F}$ of bijective selectors of $F$ such that $|\mathcal{F}|<\kappa$ and $F(x) = \{ f(x): f\in\mathcal{F}\}$ for each $x\in X$. We apply this result to $G$-space representations of balleans.