  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.
