INTRODUCTION

Summary

It is my hope that the methods developed in this text will lead to an interesting embedding of algebraic topology in a purely algebraic category, namely, some category of partially ordered rings. At the same time, the theory provides a convenient abstract setting for the theory of real semi-algebraic sets, quite analogous to commutative algebra as a setting for modern algebraic geometry.

I might motivate the study of partially ordered rings (somewhat frivolously) as follows. One observes that the integers, together with their ordering, is an initial object for a lot of mathematics. On the one hand, consideration of order properties leads to the topology of the real line, then to Euclidean spaces, and eventually to abstract continuity and general point set topology. On the other hand, consideration of arithmetic properties leads to the abstract theory of rings, fields, ideals, and modules.

Following either route, one can go too far. Completely general topological spaces and continuous maps are uninteresting. Completely general rings and modules are uninteresting. Thus the mainstream in topology concentrates on nice spaces (for example, polyhedra and manifolds) and the mainstream in algebra concentrates on nice rings (for example, finitely generated rings over fields, subrings of the complex numbers and their homomorphic images.) The two theories seem to intersect eventually in category theory and semi-simplicial homotopy theory. The topologists put back in some algebra and the algebraists put back in some topology.