Spaces of orderings and Witt rings of planar ternary rings
The author translates part of the theory of ordered fields into the context of ordered projective planes which enables him to apply Marshall's theory of spaces of orderings [\textit{M. Marshall}, Can. J. Math. 31, 320-330 (1979; Zbl 0412.10012)]. As a main result, he proves that the set of orderings of an arbitrary orderable planar ternary ring (thus the set of orderings of an arbitrary orderable projective or affine plane) forms a space of orderings in the sense of Marshall. The first section contains the basic definitions and results. The author describes the subsets H(T) and Q(T) which are the equivalents to \(K^{*2}\) and resp. \(\Sigma K^{*2}\) in the theory of ordered fields K. A planar ternary ring (shortly PTR) T is called formally-real iff 0 is not a finite sum of elements of H(T) and equivalently iff T is orderable (1.2). Then he introduces preorderings and orderings using the standard axioms. The orderings correspond 1-1 to the orders which are certain quadratic characters of \(T^*\) fulfilling axioms similar to the ones of signature functions in formally-real fields (1.3). This leads to the equivalence between ``formally-real'' and the existence of a preordering on T (which always extends to an ordering). These results allow the author to bring PTR's into the context of Marshall's abstract theory of forms. The main result of section 2 is Theorem 2.3: If T is a formally-real PTR and P a preordering of T, then \((X/P,T^*/P)\) is a space of orderings. The proof makes use of the simplified set of axioms for spaces of signatures (resp. orderings) which were introduced by \textit{M. Marshall} [J. Pure Appl. Algebra 57, No.2, 159-164 (1989; Zbl 0668.10027)]. Theorem 2.5 states that this does not yield any ``new'' finite spaces of orderings, as any finite space of orderings of a formally-real PTR already is isomorphic to the space of orderings of some Pythagorean commutative field. Section 3 deals with the abstract (reduced) Witt rings of PTR's. The author first establishes a representation criterion (3.1). Similarly he proves chain equivalence which then leads to the description of the Witt ring \(W_ P(T)\) as a residue class ring of an integral group ring, thereby extending a classical description of the reduced Witt ring into this context.
scientific article; zbMATH DE number 4112688
scientific article