Trichotomy for positive cones and a maximality counterexample

TL;DR

This paper extends the theory of positive cones on algebras with involution, showing their order relations are nearly total and providing a counterexample of a non-maximal prepositive cone, thus deepening the understanding of their structure.

Contribution

It demonstrates that positive cones induce nearly total order relations and constructs a maximality counterexample for prepositive cones, advancing the algebraic theory of involutions.

Findings

Positive cones induce partial orders close to total orders.

Positive cones are maximal prepositive cones.

Existence of non-maximal prepositive cones.

Abstract

In [4] we developed the theory of positive cones on finite-dimensional simple algebras with involution, inspired by the classical Artin-Schreier theory of orderings on fields, and based on the notion of signatures of hermitian forms [1]. In a subsequent paper [3], we developed the associated "valuation theory", based on Tignol-Wadsworth gauges [7, 8, 9]. In this short note, we present the following two additional results: (1) Whereas positive cones on fields correspond to total order relations, positive cones on algebras with involution only give rise to partial order relations. We show that the order relation defined by a positive cone is as close to total as possible, cf. Theorem 2.5. (2) Positive cones are maximal prepositive cones, which begs the question if there are prepositive cones that are not maximal. We answer this question in the affirmative in Section 3, using techniques that illustrate the interplay between positive cones and gauges.