Cantor-Schröder-Bernstein theorem for a class of countable linear orders

Abstract

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

9 pages, 1 figure; ORCID ID corrected