Classifying equivalence relations in the Ershov hierarchy

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.