Archive for Mathematical Logic

, Volume 58, Issue 5–6, pp 543–563 | Cite as

Degrees of bi-embeddable categoricity of equivalence structures

  • Nikolay Bazhenov
  • Ekaterina Fokina
  • Dino RosseggerEmail author
  • Luca San Mauro
Open Access


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.


Computable categoricity Bi-embeddability Degrees of categoricity Degrees of bi-embeddable categoricity 

Mathematics Subject Classification




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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© The Author(s) 2018

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Sobolev Institute of MathematicsNovosibirskRussia
  2. 2.Novosibirsk State UniversityNovosibirskRussia
  3. 3.Institute of Discrete Mathematics and GeometryVienna University of TechnologyWienAustria

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