Ordered Join Geometries
In this chapter, as in the previous one, a new postulate is introduced. The postulate is equivalent to the familiar Euclidean property of linear order: If three distinct points are collinear, then one of the three is between the other two. Join geometries which satisfy the new postulate are called ordered join geometries or ordered geometries. Ordered geometries are exchange geometries studied in the last chapter, but the results there will not be used in the present investigation. A flood of results is produced. First come formulas for lines, rays and segments, expressing how they are divided into subrays and subsegments by their points. Next come many properties of polytopes familiar in Euclidean geometry and easily accessible to intuition. Then follow properties of convex sets, less familiar in classical geometry but no less important, the theorems of Radon, Helly and Caratheodory and related results. These flow from a sharpened expansion formula for the linear hull of a finite set mediated by a new type of dependence of points. This new type of dependence is defined in terms of the convex hull operation and is called convex dependence. It implies linear dependence in any join geometry and is equivalent to linear dependence in an ordered geometry. Finally, separation of linear spaces by linear subspaces is studied, and results familiar in Euclidean geometry are obtained for ordered geometries.
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