Dimension Theory in Dense Regular Groups

<p>A dense regular group is a densely ordered group elementarily equivalent to a subgroup of R. If a dense regular group is divisible then it is o-minimal. In this thesis we examine the structure of the definable sets in dense regular groups which may not be divisible, but have only finitely many congruence classes mod <em>n</em> for every positive integer <em>n</em>. We define an appropriate notion of "cell," prove a cell-decomposition theorem similar to that which is known for o-minimal structures, and use cells to develop a dimension theory for the definable sets. In 2009 Zilber proposed a set of axioms which a dimension theory on a <em>topological</em> structure might satisfy, and he can show that these axioms lead to a quantifier elimination result. These axioms apply to R, Q<em>_p</em>, and every o-minimal structure with their standard topologies. We define a natural topology for the groups under investigation here—one in which the definable closed sets have positive quantifier-free definitions—and show that the dimension theory defined here satisfies all of Zilber's axioms.</p>