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A Theory Of Qualitative Similarity

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  • Klaus Nehring

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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  • Klaus Nehring, "undated". "A Theory Of Qualitative Similarity," Department of Economics 97-10, California Davis - Department of Economics.
    • Handle: RePEc:fth:caldec:97-10
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    File URL: http://www.econ.ucdavis.edu/working_papers/97-10.pdf
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    Cited by:

    1. Bervoets, Sebastian & Gravel, Nicolas, 2007. "Appraising diversity with an ordinal notion of similarity: An axiomatic approach," Mathematical Social Sciences, Elsevier, vol. 53(3), pages 259-273, May.

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