Classes of Ulm type and coding rank-homogeneous trees in other structures

Journal of Symbolic Logic 76 (3):846 - 869 (2011)
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Abstract

The first main result isolates some conditions which fail for the class of graphs and hold for the class of Abelian p-groups, the class of Abelian torsion groups, and the special class of "rank-homogeneous" trees. We consider these conditions as a possible definition of what it means for a class of structures to have "Ulm type". The result says that there can be no Turing computable embedding of a class not of Ulm type into one of Ulm type. We apply this result to show that there is no Turing computable embedding of the class of graphs into the class of "rank-homogeneous" trees. The second main result says that there is a Turing computable embedding of the class of rank-homogeneous trees into the class of torsion-free Abelian groups. The third main result says that there is a "rank-preserving" Turing computable embedding of the class of rank-homogeneous trees into the class of Boolean algebras. Using this result, we show that there is a computable Boolean algebra of Scott rank ${\mathrm{\omega }}_{1}^{\mathrm{C}\mathrm{K}}$

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Alexander Melnikov
National Research University Higher School of Economics

Citations of this work

Computable Abelian groups.Alexander G. Melnikov - 2014 - Bulletin of Symbolic Logic 20 (3):315-356,.
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New Degree Spectra of Abelian Groups.Alexander G. Melnikov - 2017 - Notre Dame Journal of Formal Logic 58 (4):507-525.
Strange Structures from Computable Model Theory.Howard Becker - 2017 - Notre Dame Journal of Formal Logic 58 (1):97-105.
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References found in this work

Computable structures of rank.J. F. Knight & J. Millar - 2010 - Journal of Mathematical Logic 10 (1):31-43.
Model Theory: An Introduction.David Marker - 2003 - Bulletin of Symbolic Logic 9 (3):408-409.
Scott sentences and admissible sets.Mark Nadel - 1974 - Annals of Mathematical Logic 7 (2):267.

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