2014/09/27 by Matthew Harrison‐Trainor, Alexander Melnikov, Harrison-Trainor, Matthew +3
Computer Science · Mathematics · #03C57 #03D45 #12Y05 #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #FOS: Mathematics #Logic (math.LO) #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.1409.7747
openalex publication_date 2014/09/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We give a sufficient condition for an algebraic structure to have a computable presentation with a computable basis and a computable presentation with no computable basis. We apply the condition to differentially closed, real closed, and difference closed fields with the relevant notions of independence. To cover these classes of structures we introduce a new technique of safe extensions that was not necessary for the previously known results of this kind. We will then apply our techniques to derive new corollaries on the number of computable presentations of these structures. The condition also implies classical and new results on vector spaces, algebraically closed fields, torsion-free abelian groups and Archimedean ordered abelian groups.