[Paper Review] Fans in the Theory of Real Semigroups I. Algebraic Theory

Max Dickmann, Alejandro Petrovich|arXiv (Cornell University)|Mar 21, 2017

Rings, Modules, and Algebras8 references19 citations

TL;DR

This paper introduces and develops the algebraic theory of fans in real semigroups (RS-fans), a new class of real semigroups that generalize fans from field theory, order spaces, and special groups. By leveraging the underlying ternary semigroup structures and duality with abstract real spectra, the authors establish foundational properties of RS-fans, including their convexity and prime ideal structure, laying the groundwork for a categorical duality in real algebraic geometry.

ABSTRACT

In a previous paper we introduced the notion of a {\it real semigroup} (RS) as an axiomatic framework to study diagonal quadratic forms with arbitrary entries over (commutative, unitary) semi-real rings. Two important classes of RSs were studied at length in previous papers. In this paper we introduce and develop the algebraic theory of {\it RS-fans}, a third class of RSs providing a vast generalization of homonymous notions previously existing in field theory and in the theories of abstract order spaces and of reduced special groups; for a background on fans, see paragraph A of the Introduction, below. The contents of this paper are briefly reviewed in paragraph B of the Introduction. The combinatorial theory of the structures dual to RS-fans, called {\it ARS-fans}, is the subject of the paper: M. Dickmann, A. Petrovich, {\em Fans in the Theory of Real Semigroups. II. Combinatorial Theory}, 20 pp., submitted., a continuation of the present paper.

Motivation & Objective

• To develop a general, intrinsic notion of fans in the category of real semigroups (RSs), extending prior notions from fields, abstract order spaces, and reduced special groups.
• To establish the algebraic theory of RS-fans using the underlying ternary semigroup (TS) structures, which are fundamental to real semigroups.
• To provide a categorical duality between RS-fans and their duals in the category of abstract real spectra (ARSs), generalizing known dualities in real algebra.
• To clarify the role of fans in real semigroups by proving key structural properties, such as convexity and prime ideal behavior, under the ternary semigroup framework.

Proposed method

• The authors define a fan in the category of real semigroups by generalizing two equivalent characterizations from AOS and RSG: (1) the set of all homomorphisms to {±1} with h(−1) = −1, and (2) closure under triple products.
• They use the ternary semigroup (TS) structure underlying real semigroups as the algebraic foundation, replacing group-theoretic notions with ternary operations.
• The theory is built via axiomatic extension of real semigroups, incorporating a condition [Z] that ensures compatibility with the RS structure.
• The paper establishes that every RS-fan is a real semigroup satisfying specific convexity and support conditions, using the duality between RSs and ARSs.
• Key techniques include the analysis of T-convex prime ideals and their behavior under intersections of total preorders T₀ and T₁.
• The authors prove that every T-convex prime ideal in a real semigroup is simultaneously T₀- and T₁-convex, which is essential for the structure theory of fans.

Experimental results

Research questions

• RQ1 How can the notion of a fan be generalized from fields and special groups to the category of real semigroups in a way that preserves duality and intrinsic algebraic structure?
• RQ2 What are the necessary and sufficient conditions for a ternary semigroup to support a structure of a real semigroup with a fan, and how does this relate to the condition [Z]?
• RQ3 How do T-convex prime ideals behave in the context of RS-fans, and what role do they play in the duality with abstract real spectra?
• RQ4 Can the combinatorial properties of fans in AOS and RSG be lifted to the more general setting of real semigroups using ternary semigroups as the underlying algebraic framework?
• RQ5 What is the relationship between the support of a real spectrum point and the convexity of prime ideals in the context of RS-fans?

Key findings

• Every RS-fan is a real semigroup whose underlying ternary semigroup satisfies the condition [Z], ensuring compatibility with the RS structure.
• Every T-convex prime ideal in a real semigroup is both T₀- and T₁-convex, a result derived from the interplay between total preorders and convexity.
• The support of a real spectrum point α in an RS-fan is a prime ideal contained in α, and if t₀t₁ ∈ supp(α), then t₀ ∈ α or t₁ ∈ α, which implies t₀ ∈ α or t₁ ∈ α.
• There exist examples, such as the ring C(ℝ) with preorders T₀ and T₁ at points 0 and 1, where maximal ideals M₀ and M₁ are T-convex but incomparable, showing that inclusion is not automatic.
• The duality between RS-fans and ARS-fans is established via the correspondence between real spectrum points and supports of prime ideals, with each T-convex prime ideal arising as the support of a unique real spectrum point.