The theory of spaces of orderings developed in [13-18] is useful in studying real varieties at the local and semi-local level; e.g., see [7]. On the other hand, N. Schwartz shows in [27] that in studying global separation of connected components of real varieties, more general sorts of structure arise.
In general, the structures one is interested in can be described as follows: For any proper preordering (T) in a ring (A) (commutative with 1 ), there is associated a natural pairing (G_{7} \times C X_{T} \rightarrow{1,-1} . X_{T}) is the topological space consisting of all orderings of (A) lying over (T . C X_{T}) is the space of connected components of (X_{T}, G_{T}) is the factor group (G_{T}=\left{a \in A: a \neq_{P} 0 \forall P \in X_{T}\right} /\left{a \in A: a>{P} 0 \forall P \in X{T}\right}). One can ask about the structure of this pairing. For example, under what conditions do elements of (G_{T}) separate points in (C X_{T}) ? Under what conditions is ( (C X_{T}, G_{T}) ) a space of orderings? As explained in [27], this has application to real algebraic geometry, to the question of existence of polynomials having prescribed signs on the connected components of a real algebraic variety. 1995 Academic Press, Inc.