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Orders on computable torsion-free abelian groups
by
Sarah Pingrey
The George Washington University
Abstract: A countable group is computable if its domain is a computable set and its group theoretic operation is computable. We examine complexity of orders on a computable torsion-free abelian (hence orderable) group G, using Turing degrees as a complexity measure. There are continuum many Turing degrees and they form an upper semilattice under Turing reducibility. All computable sets have Turing degree zero. It is easy to see that if G is of rank 1, then G has exactly two orders and they are computable. Solomon showed that if G has a finite rank greater than 1, then G has an order in every Turing degree. On the other hand, if G is of infinite rank, then G does not necessarily have a computable order, as shown by Downey and Kurtz.
Date received: December 4, 2007
Copyright © 2007 by the author(s). The author(s) of this work and the organizers of the conference have granted their consent to include this abstract in Topology Atlas. Document # cavo-16.