Effective Embeddings for Pairs of Structures

We study computable embeddings for pairs of structures, i.e. for classes containing precisely two non-isomorphic structures. We show that computable embeddings induce a non-trivial degree structure for two-element classes consisting of computable structures, in particular the pair of linear orders \(\{\omega , \omega ^\star \}\), which are the order types of the positive integers and the negative integers, respectively.