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Computable categoricity versus relative computable categoricity

Rodney G. Downey, Asher M. Kach, Steffen Lempp, Daniel D. Turetsky (2013)

Fundamenta Mathematicae

We study the notion of computable categoricity of computable structures, comparing it especially to the notion of relative computable categoricity and its relativizations. We show that every 1 decidable computably categorical structure is relatively Δ⁰₂ categorical. We study the complexity of various index sets associated with computable categoricity and relative computable categoricity. We also introduce and study a variation of relative computable categoricity, comparing it to both computable...

Computable structures and operations on the space of continuous functions

Alexander G. Melnikov, Keng Meng Ng (2016)

Fundamenta Mathematicae

We use ideas and machinery of effective algebra to investigate computable structures on the space C[0,1] of continuous functions on the unit interval. We show that (C[0,1],sup) has infinitely many computable structures non-equivalent up to a computable isometry. We also investigate if the usual operations on C[0,1] are necessarily computable in every computable structure on C[0,1]. Among other results, we show that there is a computable structure on C[0,1] which computes + and the scalar multiplication,...

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