Effectively Existentially-Atomic Structures

The notions we study in this paper, those of existentially-atomic structure and effectively existentially-atomic structure, are not really new. The objective of this paper is to single them out, survey their properties from a computability-theoretic viewpoint, and prove a few new results about them. These structures are the simplest ones around, and for that reason alone, it is worth analyzing them. As we will see, they are the simplest ones in terms of how complicated it is to find isomorphisms between different copies and in terms of the complexity of their descriptions. Despite their simplicity, they are very general in the following sense: every structure is existentially atomic if one takes enough jumps, the number of jumps being (essentially) the Scott rank of the structure. That balance between simplicity and generality is what makes them important. Existentially atomic structures are nothing more than atomic structures, as in model theory, except that the generating formulas for the principal types are required to be existential. They were analyzed by Simmons in [Sim76, Section 2] who calls them ∃1-atomic, or strongly existentially closed. Simmons referred to [Pou72] as an earlier occurrence of these structures in the literature. Here is the formal definition: