Degrees of bi-embeddable categoricity of equivalence structures

Abstract

We study the algorithmic complexity of embeddings between bi-embeddable equivalence structures. We define the notions of computable bi-embeddable categoricity, (relative) ${\Delta }_{\alpha }^0$ 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 ${\Delta }_{\alpha }^0$ bi-embeddable categoricity and relative ${\Delta }_{\alpha }^0$ 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 $0,0^{\prime }$, or $0^{\prime \prime }$. We obtain results on index sets of computable equivalence structure with respect to bi-embeddability.

18 pages

Cited by in corpus (2)

• Degrees of bi-embeddable categoricity
• On bi-embeddable categoricity of algebraic structures