Degrees of bi-embeddable categoricity of equivalence structures

We study the algorithmic complexity of embeddings between bi-embeddable equivalence structures. We define the notions of computable bi-embeddable categoricity, (relative) Δ α 0 bi-embeddable categoricity, and degrees of bi-embeddable categoricity. These notions mirror the classical notions used to s...

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Bibliographic Details
Published in:Archive for mathematical logic 2019-08, Vol.58 (5-6), p.543-563
Main Authors: Bazhenov, Nikolay, Fokina, Ekaterina, Rossegger, Dino, San Mauro, Luca
Format: Article
Language:English
Subjects:
Algebra
Complexity
Equivalence
Isomorphism
Mathematical Logic and Foundations
Mathematics
Mathematics and Statistics
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Summary:We study the algorithmic complexity of embeddings between bi-embeddable equivalence structures. We define the notions of computable bi-embeddable categoricity, (relative) Δ α 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 Δ α 0 bi-embeddable categoricity and relative Δ α 0 bi-embeddable categoricity coincide for equivalence structures for α = 1 , 2 , 3 . We also prove that computable equivalence structures have degree of bi-embeddable categoricity 0 , 0 ′ , or 0 ′ ′ . We furthermore obtain results on the index set complexity of computable equivalence structure with respect to bi-embeddability.
ISSN:0933-5846
1432-0665
DOI:10.1007/s00153-018-0650-3