[Paper Review] Abstract Vergleichsstellensätze for preordered semifields and semirings I

T. A. Fritz|arXiv (Cornell University)|Mar 30, 2020

Fuzzy Logic and Control Systems4 citations

TL;DR

This paper establishes two abstract Vergleichsstellensätze for preordered semifields and semirings, proving that the preorder on a semifield is the intersection of all total extensions, and deriving a non-Archimedean generalization of Strassen’s and Krivine–Kadison–Dubois’ Positivstellensätze using homomorphisms to the tropical reals. The results unify and extend foundational separation theorems in real algebra beyond fields and rings.

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.

Motivation & Objective

• To develop foundational tools for real algebra in the context of preordered semifields and semirings, which lack additive inverses and are central in applications to probability, graph theory, and complexity theory.
• To generalize classical separation theorems (Positivstellensätze) to non-Archimedean, non-field settings where the positive cone cannot characterize the preorder.
• To establish a Vergleichsstellensatz for preordered semifields showing that the preorder equals the intersection of all total semifield preorders extending it.
• To derive a non-Archimedean Vergleichsstellensatz for preordered semirings using homomorphisms to the tropical reals, recovering known results as special cases.

Proposed method

• Prove that every preordered semifield preorder is the intersection of its total extensions via a novel polynomial identity (Lemma 5.1), generalizing the classical field result.
• Use order cancellativity and quasi-complements to reduce the problem to a setting where Strassen’s Vergleichsstellensatz applies.
• Employ a telescoping argument with geometric series in a subsemiring to show that $ x \leq y + \varepsilon $ for all $ \varepsilon > 0 $, implying $ x \leq y $ under strict inequality conditions.
• Characterize multiplicatively Archimedean totally preordered semifields as order-embeddable into $ \mathbb{R}_+ $, $ \mathbb{R}_+^{\mathrm{op}} $, $ \mathbb{TR}_+ $, or $ \mathbb{TR}_+^{\mathrm{op}} $.
• Apply the main theorems to recover the Krivine–Kadison–Dubois Positivstellensatz via a reduction to Strassen’s result in the Archimedean case.
• Use the compactness of the space of tropical spectrum $ \mathsf{TSper}(S) $ to strengthen inequalities from $ \phi(x) \leq \phi(y) $ to $ \phi(x + \varepsilon) < \phi(y) $, enabling cancellation.

Experimental results

Research questions

• RQ1 How can separation theorems (Vergleichsstellensätze) be generalized from preordered rings and fields to preordered semifields and semirings?
• RQ2 What is the role of total extensions in characterizing preorders on semifields without additive inverses?
• RQ3 Can Strassen’s Vergleichsstellensatz be extended to non-Archimedean preordered semirings using tropical reals as targets?
• RQ4 How does the absence of a positive cone in semifields necessitate a different form of separation theorem compared to classical Positivstellensätze?
• RQ5 To what extent can the Krivine–Kadison–Dubois Positivstellensatz be derived from a more general Vergleichsstellensatz in the semiring setting?

Key findings

• The preorder on any preordered semifield is the intersection of all total semifield preorders extending it, establishing a fundamental structural result.
• Every multiplicatively Archimedean totally preordered semifield order-embeds into one of $ \mathbb{R}_+ $, $ \mathbb{R}_+^{\mathrm{op}} $, $ \mathbb{TR}_+ $, or $ \mathbb{TR}_+^{\mathrm{op}} $, generalizing the classical Archimedean embedding into $ \mathbb{R} $.
• A non-Archimedean Vergleichsstellensatz is established for preordered semirings using homomorphisms to the tropical reals, recovering Strassen’s result as a special case.
• The Krivine–Kadison–Dubois Positivstellensatz is derived as a corollary from the main theorem, showing that the classical result is a special case of the abstract framework.
• The proof technique relies on a novel polynomial identity (Lemma 5.1) and a telescoping argument with geometric series in a quasi-complemented, order-cancellative subsemiring.
• The space of tropical spectrum $ \mathsf{TSper}(S) $ is used to derive strict inequalities, enabling order cancellation and proving $ x \leq y $ from $ \phi(x) < \phi(y) $ for all $ \phi $.