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A THEORY OF QUALITATIVE SIMILARITY

Klaus Nehring

Department of Economics from California Davis - Department of Economics

Abstract: The central result of this paper establishes an isomorphism between two types of mathematical structures: "ternary preorders" and "convex topologies." The former are characterized by reflexivity, symmetry and transitivity conditions, and can be interpreted geometrically as ordered betweenness relations; the latter are defined as intersection-closed families of sets satisfying an "abstract convexity" property. A large range of examples is given. As corollaries of the main result we obtain a version of Birkhoff's representation theorem for finite distributive lattices, and a qualitative version of the representation of ultrametric distances by indexed taxonomic hierarchies.

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