Quotients of Index Two and General Quotients in a Space of Orderings

Abstract

In this paper we investigate quotient structures and quotient spaces of a space of orderings arising from subgroups of index two. We provide necessary and sufficient conditions for a quotient structure to be a quotient space that, among other things, depend on the stability index of the given space. The case of the space of orderings of the field $\mathbb{Q}(x)$ is particularly important, since then the theory developed simplifies significantly. A part of the theory firstly developed for quotients of index $2$ generalizes in an elegant way to quotients of index $2^n$ for arbitrary finite n. Numerous examples are provided.