On Ordered Geometries

Abstract

In Theorem 2.20 of his Geometric Algebra, Artin shows that any ordering of a plane geometry is equivalent to a weak ordering of its skew field. Referring to his Theorem 1. 16 that every weakly ordered field with more than two elements is ordered, he deduces his Theorem 2.21 that any ordering of a Desarguian plane with more than four points is (canonically) equivalent to an ordering of its field. We should like to present another proof of this theorem stimulated by Lipman's paper [this Bulletin, vol.4, 3, pp. 265-278]. Our proof seems to bypass Artin's Theorem 1. 16; cf. the postscript.

Ordered Desarguesian Affine Hjelmslev Planes

“…r QO -r OP and r co = T OB ), then (P, B, Q).Proof. This is very similar to ([4], p. 31).4.3. LEMMA.…”

“…Also (JB, A, C) and (B, A, D) imply C, D | A (cf. [4]). Clearly for the parallel projection of 3.2. if A, B, C I I and BC | A, then B'C | A'.…”

“…It can easily be verified, using 2.1, that our definition is independent of the choices of O and A (cf. [4]). Thus we may write H + (0, A) = H + .…”

“…[3]). In his paper on ordered geometries [4], P. Scherk discussed the equivalence of an ordering of a Desarguesian affine plane with an ordering of its coordinatizing division ring. We shall define an ordered D.A.H.…”

Ordered Line and Skew-Fields in the Desargues Affine Plane

“…He shows that that any ordering of a Desargues plane with more than four points is (canonically) equivalent to an ordering of its field. In his paper on ordered geometries [14], P. Scherk considers the equivalence of an ordering of a Desarguesian affine plane with an ordering of its coordinatizing division ring. Considerable work on ordered plane geometries has been done (see, e.g., J. Lipman [9], V.H.…”