Abstract Vergleichsstellensätze for preordered semifields and semirings I

Abstract

Real algebra is usually thought of as the study of certain kinds of preorders on fields and rings. Among its core themes are the separation theorems known as Positivstellensätze. However, there is a nascent subfield of real algebra which studies preordered semirings and semifields, which is motivated by applications to probability, graph theory and theoretical computer science, among others. Here, we contribute to this subfield by developing a number of foundational results for it, with two abstract Vergleichsstellensätze being our main theorems. Our first Vergleichsstellensatz states that every semifield preorder is the intersection of its total extensions. We apply this to derive our second main result, a Vergleichsstellensatz for certain non-Archimedean preordered semirings in which the homomorphisms to the tropical reals play an important role. We show how this result recovers the existing Vergleichsstellensatz of Strassen and (through the latter) the classical Positivstellensatz of Krivine--Kadison--Dubois.

47 pages. v3: substantial revision of part of earlier version. Other parts will appear separately. v4, v5: minor revision

Cited by in corpus (6)

• Matrix majorization in large samples
• Matrix Majorization in Large Samples with Varying Support Restrictions
• Dagger categories and the complex numbers: Axioms for the category of finite-dimensional Hilbert spaces and linear contractions
• Conditions for Large-Sample Majorization of Pairs of Flat States in Terms of $\alpha$-z Relative Entropies
• Asymptotic and catalytic containment of representations of $\mathsf{SU}(n)$
• Characterizing the asymptotic and catalytic stochastic orders on topological abelian groups