Algebraic and elementary properties of Rogers semilattices and Reverse mathematics

  • Mustafa, Manat (PI)
  • Tleuliyeva, Zhansaya (Other participant)
  • Kamet, Madina (Other participant)
  • Tursynov, Adil (Other participant)

Project: FDCRGP

Project Details

Grant Program

Faculty Development Competitive Research Grant Program (General) 2024-2026

Project Description

The project explores uniform enumeration procedures for elements of a family of sets, facilitated by computable numberings. The complexity between two numberings can be measured by their ability to transform one enumeration procedure into another. The Rogers semilattice of a family characterizes the relative complexity of all its computable numberings. The concept of computable numberings aligns with the idea of computability within the hierarchical class of sets to which the family belongs. Various hierarchies, such as arithmetical, analytical, and Feiner hierarchies, rely on oracle computations for computability, while the Ershov hierarchy utilizes approximations of the limiting classical computations.
The overarching objective is to explore the shared and distinct properties of known computability concepts. The approach involves studying the algebraic and elementary properties of Rogers semilattices in different hierarchies, aiming to identify global invariants that highlight significant distinctions in the algebraic characteristics of the Rogers semilattices belonging to arithmetical, analytical, Ershov's, and Feiner's hierarchies. These invariants encompass cardinality, minimal and maximal elements, elementary theories, and types of isomorphism.
Moreover, the project aims to analyze Rogers semilattices from the perspective of reverse mathematics
StatusActive
Effective start/end date1/1/24 → 12/31/26

Keywords

  • Computability theory
  • Computable numberings
  • Rogers semilattices
  • Punctual Structure
  • Reverse Mathematics
  • Computable Structures

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  • 2 Conference contribution
  • 1 Article

  • Free bicommutative superalgebras

    Drensky, V., Ismailov, N., Mustafa, M. & Zhakhayev, B., Aug 15 2024, In: Journal of Algebra. 652, p. 158-187 30 p.

    Research output: Contribution to journal › Article › peer-review

    Open Access
    1 Citation (Scopus)

  • On Arithmetical Numberings in Reverse Mathematics

    Bazhenov, N., Fiori-Carones, M. & Mustafa, M., 2024, Twenty Years of Theoretical and Practical Synergies - 20th Conference on Computability in Europe, CiE 2024, Proceedings. Levy Patey, L., Pimentel, E., Galeotti, L. & Manea, F. (eds.). Springer Science and Business Media Deutschland GmbH, p. 126-138 13 p. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); vol. 14773 LNCS).

    Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

  • On Learning Families of Ideals in Lattices and Boolean Algebras

    Bazhenov, N. & Mustafa, M., 2024, Theory and Applications of Models of Computation - 18th Annual Conference, TAMC 2024, Proceedings. Chen, X. & Li, B. (eds.). Springer Science and Business Media Deutschland GmbH, p. 1-13 13 p. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); vol. 14637 LNCS).

    Research output: Chapter in Book/Report/Conference proceeding › Conference contribution