## Abstract

Given a structure ℳ over ω and a syntactic complexity class E, we say that a subset is E-definable in ℳ if there exists a C-formula Θ(x) in the language of ℳ such that for all x ∈ ω, we have x ∈ A iff Θ(x) is true in the structure. S. S. Goncharov and N. T. Kogabaev [Vestnik NGU, Mat., Mekh., Inf., 8, No. 4, 23-32 (2008)] generalized an idea proposed by Friedberg [J. Symb. Log., 23, No. 3, 309-316 (1958)], introducing the notion of a E-classification of M: a computable list of E-formulas such that every E-definable subset is defined by a unique formula in the list. We study the connections amongΣ10−, d−Σ10−, and Σ20-classifications in the context of two families of structures, unbounded computable equivalence structures and unbounded computable injection structures. It is stated that every such injection structure has a Σ10−classification, a Σ10−classification, and a Σ20-classification. In equivalence structures, on the other hand, we find a richer variety of possibilities.

Original language | English |
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Pages (from-to) | 383-404 |

Number of pages | 22 |

Journal | Algebra and Logic |

Volume | 58 |

Issue number | 5 |

DOIs | |

Publication status | Published - 1 Nov 2019 |

### Bibliographical note

Publisher Copyright:© 2019, Springer Science+Business Media, LLC, part of Springer Nature.

## Keywords

- d−Σ -classification
- unbounded computable equivalence structure
- unbounded computable injection structure
- Σ -classification

## ASJC Scopus subject areas

- Analysis
- Algebra and Number Theory
- Logic