Torsion-free abelian groups with optimal Scott families

We prove that for any computable successor ordinal of the formα = δ + 2k(δ limit andk ∈ ω) there exists computable torsion-free abelian group(TFAG) that is relativelyΔα0-categorical and notΔα−10-categorical. Equivalently, for any such α there exists a computable TFAG whose initial segments are uniformly described by Σαc infinitary computable formulae up to automorphism (i.e. it has a c.e. uniformly Σαc-Scott family), and there is no syntactically simpler (c.e.) family of formulae that would capture these orbits. As far as we know, the problem of finding such optimal examples of (relatively)Δα0-categorical TFAGs for arbitrarily large α was first raised by Goncharov at least 10 years ago, but it has resisted solution (see e.g. Problem 7.1 in survey [Computable abelian groups, Bull. Symbolic Logic 20(3) (2014) 315–356]). As a byproduct of the proof, we introduce an effective functor that transforms a 0″-computable worthy labeled tree (to be defined) into a computable TFAG. We expect that this technical resul...

[1]  Rodney G. Downey,et al.  Effectively categorical abelian groups , 2013 .

[2]  Joaquín Pascual,et al.  Infinite Abelian Groups , 1970 .

[3]  G. Higman.  Subgroups of finitely presented groups , 1961, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[4]  B. L. Waerden.  Eine Bemerkung über die Unzerlegbarkeit von Polynomen , 1930 .

[5]  R. Baer,et al.  Abelian groups without elements of finite order , 1937 .

[6]  Rodney G. Downey,et al.  THE ISOMORPHISM PROBLEM FOR TORSION-FREE ABELIAN GROUPS IS ANALYTIC COMPLETE. , 2008 .

[7]  C. Ash,et al.  Recursive labelling systems and stability of recursive structures in hyperarithmetical degrees , 1986 .

[8]  N. G. Khisamiev,et al.  Chapter 17 Constructive abelian groups , 1998 .

[9]  Julia F. Knight,et al.  Classes of Ulm type and coding rank-homogeneous trees in other structures , 2011, The Journal of Symbolic Logic.

[10]  S. Goncharov,et al.  Computable Structure and Non-Structure Theorems , 2002 .

[11]  Julia F. Knight,et al.  Isomorphism relations on computable structures , 2012, J. Symb. Log..

[12]  A. Nerode,et al.  Effective content of field theory , 1979 .

[13]  A. I. Mal'tsev.  CONSTRUCTIVE ALGEBRAS I , 1961 .

[14]  N. A. Bazhenov.  Δ20-Categoricity of Boolean Algebras , 2014 .

[15]  R. Downey.  Chapter 14 Computability theory and linear orderings , 1998 .

[16]  Alexander G. Melnikov,et al.  Computable Abelian Groups , 2014, Bull. Symb. Log..

[17]  Charles F. D. McCoy.  Delta20 - categoricity in Boolean algebras and linear orderings , 2003, Ann. Pure Appl. Log..

[18]  Kyle Riggs.  The decomposability problem for torsion-free abelian groups is analytic-complete , 2013 .

[19]  Rodney G. Downey,et al.  The complexity of computable categoricity , 2015 .

[20]  Matthew Harrison-Trainor,et al.  Computable Functors and Effective interpretability , 2017, J. Symb. Log..

[21]  Jr. Hartley Rogers.  Theory of Recursive Functions and Effective Computability , 1969 .

[22]  Rodney G. Downey,et al.  Iterated effective embeddings of abelian p-groups , 2014, Int. J. Algebra Comput..

[23]  Victor A. Ocasio.  Computability in the class of Real Closed Fields , 2014 .

[24]  A. Nerode,et al.  Recursively enumerable vector spaces , 1977 .

[25]  Alexander G. Melnikov,et al.  Jump degrees of torsion-free abelian groups , 2012, The Journal of Symbolic Logic.

[26]  Asher M. Kach,et al.  Limitwise monotonic functions, sets, and degrees on computable domains , 2010, J. Symb. Log..

[27]  J. Shepherdson,et al.  Effective procedures in field theory , 1956, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[28]  Russell G. Miller,et al.  Turing degree spectra of differentially Closed Fields , 2017, J. Symb. Log..

[29]  Ewan J. Barker.  Back and Forth Relations for Reduced Abelian p-Groups , 1995, Ann. Pure Appl. Log..

[30]  Wesley Calvert,et al.  The isomorphism problem for computable Abelian p-groups of bounded length , 2004, Journal of Symbolic Logic.

[31]  Julia F. Knight,et al.  Generic Copies of Countable Structures , 1989, Ann. Pure Appl. Log..

[32]  Richard Lawrence Smith,et al.  Two theorems on autostability in p-Groups , 1981 .

[33]  M. Rabin.  Computable algebra, general theory and theory of computable fields. , 1960 .

[34]  L. Pontrjagin,et al.  The Theory of Topological Commutative Groups , 1934 .

[35]  A. V. Yakovlev.  Torsion free Abelian groups of finite rank and their direct decompositions , 1991 .

[36]  N. G. Khisamiev,et al.  Effectively totally decomposable abelian groups , 1997 .

[37]  Alexander G. Melnikov.  0"-Categorical Completely Decomposable Torsion-Free Abelian Groups , 2009, CiE.

[38]  Greg Hjorth,et al.  The isomorphism relation on countable torsion free abelian groups , 2002 .

[39]  Arkadii M. Slinko,et al.  Degree spectra and computable dimensions in algebraic structures , 2002, Ann. Pure Appl. Log..

[40]  Alexander G. Melnikov.  New Degree Spectra of Abelian Groups , 2017, Notre Dame J. Formal Log..

[41]  Paul M. Weichsel,et al.  ON p-ABELIAN GROUPS , 1967 .

[42]  A A Fomin.  TORSION-FREE ABELIAN GROUPS OF RANK 3 , 1991 .

[43]  Rodney G. Downey,et al.  Computable completely decomposable groups , 2014 .

[44]  MATTHEW HARRISON-TRAINOR,et al.  Independence in computable algebra , 2014, 1409.7747.

[45]  Rodney G. Downey,et al.  Uniformity in Computable Structure Theory , 2001 .

[46]  Julia A. Knight,et al.  Computable structures and the hyperarithmetical hierarchy , 2000 .