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We work in the big category of commutative multirings with 1. A multiring is just a ring with multivalued addition. We show that certain key results in real algebra (parts of theArtin-Schreier theory for fields and the Positivstellensatz for rings) extend to the corresponding objects in this category. We also show how the space of signs functor A Q red (A) defined in [C. ]. As a corollary we obtain a first-order description of a space of signs as a multiring satisfying certain additional properties. This simplifies substantially the description given in [M. Dickmann, A. Petrovich, Real semigroups and abstract real spectra I, Cont. Math. 344 (2004) 99-119].

The paper is a continuation of work initiated by the first two authors in [K-M]. Section 1 is introductory. In Section 2 we prove a basic lemma, Lemma 2.1, and use it to give new proofs of key technical results of Scheiderer in [S1] [S2] in the compact case; see Corollaries 2.3, 2.4 and 2.5. Lemma 2.1 is also used in Section 3 where we continue the examination of the case n = 1 initiated in [K-M], concentrating on the compact case. In Section 4 we prove certain uniform degree bounds for representations in the case n = 1, which we then use in Section 5 to prove that (‡) holds for basic closed semi-algebraic subsets of cylinders with compact cross-section, provided the generators satisfy certain conditions; see Theorem 5.3 and Corollary 5.5. Theorem 5.3 provides a partial answer to a question raised by Schmüdgen in [Sc2]. We also show that, for basic closed semi-algebraic subsets of cylinders with compact cross-section, the sufficient conditions for (SMP) given in [Sc2] are also necessary; see Corollary 5.2(b). In Section 6 we prove a module variant of the result in [Sc2], in the same spirit as Putinar's variant [Pu] of the result in [Sc1] in the compact case; see Theorem 6.1. We apply this to basic closed semi-algebraic subsets of cylinders with compact cross-section; see Corollary 6.4. In Section 7 we apply the results from Section 5 to solve two of the open problems listed in [K-M]; see Corollary 7.1 and Corollary 7.4. In Section 8 we consider a number of examples in the plane. In Section 9 we list some open problems.

1. Introduction. A space of orderings will refer to what was called a “set of quasi-orderings” in [5]. That is, a space of orderings is a pair (X, G) where G is an elementary 2-group (i.e. x2 = 1 for all x ∈ G) with a distinguished element – 1 ∈ G, and X is a subset of the character group x(G) = Horn (G, {1, –1};) satisfying the following properties:01: X is a closed subset of χ(G).02: σ(−l) = −1 holds for all σ ∊ X.03: X⊥ = {a ∊ G|χa = 1 for all a ∊ X} = 1.04: If f and g are forms over G and if x ∊ Df⊗g, then there exist y ∊ Df and z ∊ Dg such that x ∊ D(y, z).