The Operation of Extension

Abstract

In this chapter the theory of join operations is expanded by defining an operation of extension, which is a sort of inverse to join. Three postulates involving extension are introduced to complete the basic postulate set. These are employed to derive principles and formulas involving extension and join which supplement and enrich the formal theory of join in Chapters 2 and 3. The theory is applied to new ideas: extreme points of convex sets, linear order of points and two categories of convex set referred to as open and closed. New results are obtained on familiar ideas: Theorem 4.28—any join of points is an open convex set; Theorem 4.30—the interior of a polytope P is the join of the points of any finite set of generators of P; and Theorem 4.31 − I(AB) = I(A)I(B), provided I(A), I(B) ≠ Ø.KeywordsExtreme PointInterior PointDistinct PointEuclidean GeometryEuclidean PlaneThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.