AXIOMATIC CONVEXITY THEORY

WILLIAM ELLIS FENTON, Purdue University

Abstract

This paper investigates the fundamental properties of a convexity space, an axiomatic structure devised by Buchi and Mei which is similar to a matroid but incorporates the notion of opposite element. Directed graphs and vector spaces over ordered fields are models of this system. A concise presentation of Mei's dissertation is included. A prerequisite was a rigorous definition of directed graph, to allow proofs of the convexity axioms. This definition led to a characterization of the series-parallel graphs with a "battery", an edge between the designated pair of vertices, as those graphs not containing a K(,4) graph. The two basic types of convex sets are the self-opposite and the opposite-disjoint convex sets, called the flats and sharps respectively. The flats are closed under both operators of the system and are thus the subspaces. Given a flat and an element not contained in it, the flat can be extended to a maximal flat not containing this element; the same is true for sharps. Extending an arbitrary convex set with regard to an outside element yields a relatively maximal convex set, which need not be maximal. Maximal and relatively maximal convex sets contain a maximal sharp but not necessarily a maximal flat. Assuming fullness gives several strong properties: Every component of rank greater than two is dense. The space is complete (in every subspace a maximal contains a maximal flat) iff every subspace of rank two is complete, from which it is shown that a vector space over an ordered field is complete iff its field is complete in the Dedekind sense. Maximality of flats, sharps, and convex sets is preserved under relativization. Given two disjoint sharps, one can be extended to a maximal sharp disjoint from the other; this is a generalization of the Stone Separation Lemma, that (in an affine space) two disjoint convex sets can be extended to complementary convex sets.

Degree

Ph.D.

Subject Area

Mathematics

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