# Linear orders: when embeddability and epimorphism agree

## TL;DR

This paper investigates the properties of strongly surjective linear orders, establishing their descriptive set-theoretic complexity and demonstrating the existence of uncountable examples under certain hypotheses.

## Contribution

It characterizes the complexity of countable strongly surjective linear orders and proves the existence of uncountable ones beyond ZFC assumptions.

## Key findings

- Countable strongly surjective linear orders are complete for unions of analytic and coanalytic sets.
- Existence of uncountable strongly surjective orders is shown under hypotheses beyond ZFC.

## Abstract

When a linear order has an order preserving surjection onto each of its suborders we say that it is strongly surjective. We prove that the set of countable strongly surjective linear orders is complete for the class of sets which are the union of an analytic and a coanalytic set. Using hypotheses beyond ZFC, we prove the existence of uncountable strongly surjective orders.