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 nec- essary 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 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 2n for arbitrary finite n. Numerous examples are provided.