Trichotomy for positive cones and a maximality counterexample
Abstract
In [4] we developed the theory of positive cones on finitedimensional simple algebras with involution, inspired by the classical ArtinSchreier 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 TignolWadsworth 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.
 Publication:

arXiv eprints
 Pub Date:
 September 2021
 arXiv:
 arXiv:2109.09805
 Bibcode:
 2021arXiv210909805A
 Keywords:

 Mathematics  Rings and Algebras
 EPrint:
 Section 3 contains an error that invalidates the example. We think this particular example cannot be made to work at all