# Surjectively rigid chains

## TL;DR

This paper investigates the rigidity of linearly ordered sets, especially dense chains of real numbers and uncountable chains, analyzing their automorphisms and embeddings, and exploring the role of the axiom of choice.

## Contribution

It provides new insights into the rigidity properties of chains under various morphisms and introduces a Fraenkel-Mostowski model to examine the axiom of choice's impact.

## Key findings

- Rigidity properties of dense chains of real numbers analyzed.
- Uncountable dense chains of higher cardinalities studied.
- A Fraenkel-Mostowski model illustrating the axiom of choice's role provided.

## Abstract

We study rigidity properties of linearly ordered sets (chains) under automorphisms, order-embeddings, epimorphisms, and endomorphisms. We focus on two main cases, dense subchains of the real numbers, and uncountable dense chains of higher (regular) cardinalities. We also give a Fraenkel-Mostowski model which illustrates the role of the axiom of choice in one of the key proofs.