# Computable embeddings for pairs of linear orders

## TL;DR

This paper investigates computable embeddings between pairs of linear orders, revealing a non-trivial degree structure and establishing divisibility conditions for embeddings between specific pairs of linear orders.

## Contribution

It introduces the concept of computable embeddings for pairs of structures and characterizes when certain pairs of linear orders are embeddable based on divisibility.

## Key findings

- Computable embeddings induce a non-trivial degree structure for pairs of simple linear orders.
- Embeddability between specific pairs of linear orders depends on divisibility of their parameters.
- The main result characterizes embeddability of pairs of linear orders using divisibility conditions.

## Abstract

We study computable embeddings for pairs of structures, i.e. for classes containing precisely two non-isomorphic structures. Surprisingly, even for some pairs of simple linear orders, computable embeddings induce a non-trivial degree structure. Our main result shows that $\{\omega \cdot k,\omega^\star \cdot k\}$ is computably embeddable in $\{\omega \cdot t, \omega^\star \cdot t\}$ iff $k$ divides $t$.