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 furthermore obtain results on the index set complexity of computable equivalence structure with respect to bi-embeddability.
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Open access funding provided by Austrian Science Fund (FWF). The first author was supported by the Russian Foundation for Basic Research, according to the research Project No. 16-31-60058 mol_a_dk. The second, third and fourth author were supported by the Austrian Science Fund FWF through project P 27527.
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Bazhenov, N., Fokina, E., Rossegger, D. et al. Degrees of bi-embeddable categoricity of equivalence structures. Arch. Math. Logic 58, 543–563 (2019). https://doi.org/10.1007/s00153-018-0650-3
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DOI: https://doi.org/10.1007/s00153-018-0650-3