Real reduced multirings and multifields
By Martyn Marshall. 2006
64 citations recorded by the source corpus.
Abstract 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 ⇝ Q red ( A ) defined in [C. Andradas, L. Brocker, 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. Brocker, 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].
Subfields: Algebra and Number Theory, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/cf6d6c4702f6f70709cf47b52c1bce11c0966d69
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In Algebra and Number Theory
In Geometry and Topology
Citations in the corpus
- Algebraic Geometry Over Hyperrings (2015, background): “Besides from the aforementioned Connes and Consani’s approaches, M.Marshall and P.G ladki used multirings (which are more general objects than hyperrings) in relation to quadratic forms, Artin-Schreier theory, and real algebraic geometry (cf. [17], [32]).”