Effective algebraicity

Archive for Mathematical Logic 52 (1-2):91-112 (2013)
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Results of R. Miller in 2009 proved several theorems about algebraic fields and computable categoricity. Also in 2009, A. Frolov, I. Kalimullin, and R. Miller proved some results about the degree spectrum of an algebraic field when viewed as a subfield of its algebraic closure. Here, we show that the same computable categoricity results also hold for finite-branching trees under the predecessor function and for connected, finite-valence, pointed graphs, and we show that the degree spectrum results do not hold for these trees and graphs. We also offer an explanation for why the degree spectrum results distinguish these classes of structures: although all three structures are algebraic structures, the fields are what we call effectively algebraic.



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Citations of this work

Classifications of Computable Structures.Karen Lange, Russell Miller & Rebecca M. Steiner - 2018 - Notre Dame Journal of Formal Logic 59 (1):35-59.

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References found in this work

Degrees coded in jumps of orderings.Julia F. Knight - 1986 - Journal of Symbolic Logic 51 (4):1034-1042.
d-computable Categoricity for Algebraic Fields.Russell Miller - 2009 - Journal of Symbolic Logic 74 (4):1325 - 1351.
Book Reviews. [REVIEW]Wilfrid Hodges - 1997 - Studia Logica 64 (1):133-149.
Spectra of Structures and Relations.Valentina S. Harizanov & Russel G. Miller - 2007 - Journal of Symbolic Logic 72 (1):324 - 348.

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