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.
collections
- retrospective theses [1604]