Degrees of bi-embeddable categoricity of equivalence structures

TL;DR

This paper explores the computational complexity of embeddings between bi-embeddable equivalence structures, establishing the degrees of bi-embeddable categoricity and analyzing their properties.

Contribution

It introduces and compares notions of bi-embeddable categoricity, showing their equivalence for certain levels and classifying the degrees of categoricity for computable structures.

Findings

Degrees of bi-embeddable categoricity are limited to 0, 0', or 0''.

For certain levels, bi-embeddable categoricity notions coincide.

Results on index sets related to bi-embeddability in computable structures.

Abstract

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