Computable embeddings for pairs of linear orders

We study computable embeddings for pairs of structures, i.e., for classes containing precisely two nonisomorphic structures. Surprisingly, even for some pairs of simple linear orders, computable embeddings induce a nontrivial 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 $t$.