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    Linearization of an abstract convexity space

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    YongS1978m-1b.pdf (5.457Mb)

    Date

    1978

    Author

    Yong, Sin

    Degree

    Master of Science

    Discipline

    Mathematical Sciences

    Subject

    Convex sets

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    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.

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    http://knowledgecommons.lakeheadu.ca/handle/2453/2251

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