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Classifying equivalence relations in the Ershov hierarchy.

Nikolay Bazhenov1,2, Manat Mustafa3, Luca San Mauro4

  • 1Sobolev Institute of Mathematics, pr. Akad. Koptyuga 4, Novosibirsk, Russia 630090.

Archive for Mathematical Logic
|October 30, 2020
PubMed
Summary

This study explores computably enumerable equivalence relations (ceers) using a new framework. We investigate the algebraic properties and degree structures of these relations, focusing on the existence of infima and suprema.

Keywords:
Computability theoryComputably enumerable equivalence relationsErshov hierarchy

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Area of Science:

  • Computability Theory
  • Logic
  • Set Theory

Background:

  • Computably enumerable equivalence relations (ceers) are a significant area of study in computability theory.
  • The computable reducibility () is the standard tool for classifying ceers, leading to a complex degree structure.

Purpose of the Study:

  • To extend the study of c-degrees to the case.
  • To analyze the algebraic properties of the degree structure induced by on equivalence relations.
  • To investigate the existence of infima and suprema for c-degrees within this new framework.

Main Methods:

  • Utilizing the Ershov hierarchy for analysis.
  • Applying computable reducibility () to equivalence relations.
  • Proving algebraic properties of the induced degree structure.

Main Results:

  • Established several algebraic properties of the degree structure for equivalence relations.
  • Demonstrated specific findings regarding the (non)existence of infima and suprema for c-degrees.
  • Provided a foundation for further research into higher-order computability structures.

Conclusions:

  • The study successfully extends the analysis of c-degrees to the context.
  • The Ershov hierarchy proves to be a valuable tool in understanding these complex structures.
  • Further research can build upon these results to explore more intricate computability phenomena.