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Real reduced multirings and multifields. (English) Zbl 1089.14009
Summary: 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 $$A \rightsquigarrow Q_{\mathrm{red}}(A)$$ defined in [C. Andradas, L. Bröcker and J. M. Ruiz: “Constructible sets in real geometry” (1996; Zbl 0873.14044)] 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 [loc. cit.] and [M. Marshall, “Spaces of Orderings and Abstract Real Spectra”, Lect. Notes Math. 1636 (1996; Zbl 0866.12001)]. 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, in: Algebraic and arithmetic theory of quadratic forms. Proc. int. conf. Talca 2002. Contemp. Math. 344, 99-119 (2004; Zbl 1117.13026)].

##### MSC:
 14P10 Semialgebraic sets and related spaces 13J30 Real algebra 12D15 Fields related with sums of squares (formally real fields, Pythagorean fields, etc.)
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##### References:
 [1] Andradas, C.; Bröcker, L.; Ruiz, J., Constructible sets in real geometry, (1996), Springer Berlin · Zbl 0873.14044 [2] M. El Bachraoui, Relation algebras, multigroupoids, and degree, Ph.D. Thesis, University of Amsterdam, 2002. [3] Bochnak, J.; Coste, M.; Roy, M.-F., Géométrie algébrique Réelle, (1987), Springer Berlin · Zbl 0633.14016 [4] Dickmann, M.; Miraglia, F., Special groups: Boolean-theoretic methods in the theory of quadratic forms, Mem. amer. math. soc., 145, (2000) · Zbl 1052.11027 [5] Dickmann, M.; Petrovich, A., Real semigroups and abstract real spectra I, Cont. math., 344, 99-119, (2004) · Zbl 1117.13026 [6] Hochster, M., Prime ideal structure in commutative rings, Trans. amer. math. soc., 142, 43-60, (1969) · Zbl 0184.29401 [7] Marshall, M., Spaces of orderings and abstract real spectra, (1996), Springer Berlin · Zbl 0866.12001 [8] Marshall, M., Open questions in the theory of spaces of orderings, J. symbolic logic, 67, 341-352, (2002) · Zbl 1018.11017 [9] Marshall, M., The elementary type conjecture in quadratic form theory, Cont. math., 344, 275-293, (2004) · Zbl 1143.11315
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