Spaces of Orderings

Abstract

This chapter contains a fully detailed presentation of the theory of spaces of orderings. In Section 1 we reformulate the notion of space of orderings to stress the connection with the duality of topological groups and the theory of reduced quadratic forms. Section 2 is devoted to sums and extensions of spaces of orderings and their basic properties. In Section 3 we introduce spaces of finite type and their trees, which support the use of induction in many proofs. The fundamental fact that the chain length of a space of orderings is bigger than or equal to that of any subspace is proved in Section 4. Also in this section we define solid fans, impervious fans, and places, which are essential to prove in Section 5 that finite chain length is equivalent to finite type. This is the key technical result of the theory. We prove in Section 6 the local-global principle that reduces problems on forms from the whole space to its finite subspaces. In Section 7 we draw the consequences: the representation theorem, the generation formula and the stability formula. Using these, we bound each invariant s, $\bar s$, t, $\bar t$, w, l in terms of any of the others. In particular, all of them are finite or infinite at a time. The final result of the section and the chapter is a local-global separation principle.