Real reduced multirings and multifields

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 the Artin–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⇝Qred(A)A⇝Qred(A) defined in [C. Andradas, L. Bröcker, J. Ruiz, Constructible Sets in Real Geometry, Springer, Berlin, 1996; M. Marshall, Spaces of Orderings and Abstract Real Spectra, Springer, Berlin, 1996] extends to this category. The proofs are no more difficult than in the ring case. In fact they are easier. This simplifies and clarifies the presentation in [C. Andradas, L. Bröcker, J. Ruiz, Constructible Sets in Real Geometry, Springer, Berlin, 1996; M. Marshall, Spaces of Orderings and Abstract Real Spectra, Springer, Berlin, 1996]. 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].