Orderings on fields -- Spaces of orderings -- Fans and the representation theorem -- P-structures, connected components and 0 the isotropy theorem -- The real spectrum of a ring -- Abstract real spectra -- Minimal generation of constructible sets -- Structure results and realization results.

This book is of interest to students as well as experts in the area of real algebraic geometry, quadratic forms, orderings, valuations, lattice ordered groups and rings, and in model theory. The original motivation comes from orderings on fields and commutative rings. This is explained as is the important application to minimal generation of semi-algebraic sets. Many results in the new theory of abstract real spectra (also called spaces of signs) appear here for the first time. The reader needs elementary knowledge of commutative rings, ordered fields and real closed fields and valuations.