Computable reducibility of equivalence relations and an effective jump operator

TL;DR

This paper introduces a new computable jump operator for equivalence relations, demonstrating its properness and analyzing its impact on computably enumerable equivalence relations, advancing the understanding of their reducibility structure.

Contribution

It defines the computable FS-jump, proves its properness, and explores its effects on c.e. equivalence relations, extending classical jump concepts to computability theory.

Findings

The computable FS-jump is proper with respect to computable reducibility.

The jump affects the structure of c.e. equivalence relations.

Provides new tools for analyzing equivalence relation complexity.

Abstract

We introduce the computable FS-jump, an analog of the classical Friedman--Stanley jump in the context of equivalence relations on the natural numbers. We prove that the computable FS-jump is proper with respect to computable reducibility. We then study the effect of the computable FS-jump on computably enumerable equivalence relations (ceers).