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The Friedman–Stanley jump, extensively studied by descriptive set theorists, is a fundamental tool for gauging the complexity of Borel isomorphism relations. This paper focuses on a natural computable analog of this jump operator for equivalence relations on $\omega $, written ${\dotplus }$, recently introduced by Clemens, Coskey, and Krakoff. We offer a thorough analysis of the computable Friedman–Stanley jump and its connections with the hierarchy of countable equivalence relations under the computable reducibility $\leq _c$. In particular, we show that this jump gives benchmark equivalence relations going up the hyperarithmetic hierarchy and we unveil the complicated highness hierarchy that arises from ${\dotplus }$.

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).

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INVESTIGATING THE COMPUTABLE FRIEDMAN–STANLEY JUMP

COMPUTABLE REDUCIBILITY OF EQUIVALENCE RELATIONS AND AN EFFECTIVE JUMP OPERATOR

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INVESTIGATING THE COMPUTABLE FRIEDMAN–STANLEY JUMP

COMPUTABLE REDUCIBILITY OF EQUIVALENCE RELATIONS AND AN EFFECTIVE JUMP OPERATOR