Degrees of bi-embeddable categoricity of equivalence structures
Nikolay A. Bazhenov, Ekaterina Fokina, Dino Rossegger, Luca San Mauro · Archive for Mathematical Logic · 2018
We study the algorithmic complexity of embeddings between bi-embeddable equivalence structures. We define the notions of computable bi-embeddable categoricity, (relative) $$\Delta ^0_\alpha $$ bi-embeddable categoricity, and degrees of bi-embeddable categoricity. These notions mirror the classical notions used to study the complexity of isomorphisms between structures. We show that the notions of $$\Delta ^0_\alpha $$ bi-embeddable categoricity and relative $$\Delta ^0_\alpha $$ bi-embeddable categoricity coincide for equivalence structures for $$\alpha =1,2,3$$ . We also prove that computable equivalence structures have degree of bi-embeddable categoricity $$\mathbf {0},\mathbf {0}'$$ , or $$\mathbf {0}''$$ . We furthermore obtain results on the index set complexity of computable equivalence structure with respect to bi-embeddability.