Linearization of an abstract convexity space

Abstract

Axiomatic convexity space, introduced by Kay and Womble [22] , will be the main topic discussed in this thesis. An axiomatic convexity space (X,C), which is domain finite and has regular straight segments, is called a basic convexity space, A weak complete basic convexity space is a basic convexity space which is complete and has C-isomorphic property. If in addition, it is join-hull commutative then it is called (strong) complete basic convexity space. The main results presented are: a generalized line space is a weak complete basic convexity space, a complete basic convexity space is equivalent to a line space; and a complete basic convexity space whose dimension is greater than two or desarguesian and of dimension two, is a linearly open convex subset of a real affine space. Finally, we develop a linearization theory by following an approach given by Bennett [3]. A basic convexity space whose dimension is greater than two, which is join-hull commutative and has a parallelism property, is an affine space. It can be made into a vector space over an ordered division ring and the members of C are precisely the convex subsets of the vector space.