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

Published online by Cambridge University Press:  12 March 2014

E. Fokina ,
J. F. Knight ,
A. Melnikov ,
S. M. Quinn  and
C. Safranski
E. Fokina
Affiliation:
Kurt Gödel Research Center for Mathematical Logic, University of Vienna, Währinger Straße 25, 1090 Wien, Austria, E-mail: efokina@logic.univie.ac.at
J. F. Knight
Affiliation:
Department of Mathematics, 255 Hurley Hall, University of Notre Dame, Notre Dame, IN 46556-4618, USA, E-mail: knight.1@nd.edu
A. Melnikov
Affiliation:
Department of Computer Science, University of Auckland, Auckland, New Zealand, E-mail: a.melnikov@cs.auckland.ac.nz
S. M. Quinn
Affiliation:
Department of Mathematics, Dominican University, 7900 W. Division Street, River Forest, IL 60305, USA, E-mail: squinn@dom.edu
C. Safranski
Affiliation:
St. Vincent College, 300 Fraser Purchase Road, Latrobe, PA 15650, USA, E-mail: christina.safranski@email.stvincent.edu
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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 .

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 76 , Issue 3 , September 2011 , pp. 846 - 869
Copyright
Copyright © Association for Symbolic Logic 2011

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