Cantor-Schr\"oder-Bernstein theorem for a class of countable linear orders

TL;DR

This paper proves a version of the Cantor-Schr"oder-Bernstein theorem for a class of countable linear orders called shuffles, showing that mutual convex embeddability implies isomorphism.

Contribution

It establishes that any two countable shuffles embedding into each other as convex subsets are order isomorphic, extending classical order theory results.

Findings

Mutual convex embeddability implies isomorphism for countable shuffles.

Introduces a unique linear order construction called shuffle based on dense colorings.

Provides a new characterization of countable linear orders through shuffles.

Abstract

The shuffle of a non-empty countable set $ S $ of linear orders is the (unique up to isomorphism) linear order $ \Xi(S) $ obtained by fixing a coloring function $ \chi: \mathbb{Q} \to S $ having fibers dense in $ \mathbb{Q} $ and replacing each rational $ q $ in $ (\mathbb{Q}, <) $ with an isomorphic copy of $ \chi(q) $. We prove that any two countable shuffles that embed as convex subsets into each other are order isomorphic.