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Effectively closed sets and orderings on groups
by
Jennifer Chubb
George Washington University
A countable group G is computable if there is an algorithm to determine membership in G as a set, and an algorithm for multiplication on the group. G is left-orderable (bi-orderable) if there is a linear ordering of the elements of the group that is left-invarient (both left- and right-invarient). I will describe how the orderings of a countable group may be viewed as infinite paths through a binary tree, and how the orderings of a computable group correspond to paths in a computable binary tree. Taking the usual topology induced on the paths, we see that these sets are closed subsets of Cantor space, and in the computable case, we can think of them as effectively closed. The effectively closed sets have been extensively studied in computability theory, and I will describe some of the computability theoretic consequences for the spaces of orderings on groups.
Date received: December 6, 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-18.