Computable embeddings for pairs of linear orders

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 $t$.