Spaces of Signs

Spaces of signs are defined by imposing four suplementary axioms on real spaces; then, spaces of orderings are a special class of these spaces of signs. This is done in Section 1, where we also define subspaces and draw the first consequences of our definitions. Section 2 contains the fundamental properties of forms, mainly in the case of spaces of orderings. The important notion of fan is introduced in Section 3, together with its elementary properties. In Section 4 we consider local spaces of orderings and localizations, which behave very much like in ring theory. Localizations are used in Section 5 to show that the real space associated to a commutative ring with unit is actually a space of signs, and also in Section 6, to prove that the subspaces of a space of signs are again spaces of signs.