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A Characterization of Ordered Sets and Lattices via Betweenness Relations

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Abstract

For a set X with at least 3 elements, we establish a canonical one to one correspondence between all betweenness relations satisfying certain axioms and all pairs of inverse orderings “<” and “>” defined on X for which the corresponding Hasse diagram is connected and all maximal chains contain at least 3 elements. For an ordering “<”, the corresponding betweenness relation B is given by

$$B=\{(x,y,z)\in X^3\mid x<y<z {\rm \ or\ }z<y<x\}.$$

Moreover, by adding one more axiom, we obtain also a one to one correspondence between all pairs of dual lattices and all betweenness relations.

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Authors and Affiliations

  1. Fakultät für Mathematik, Technische Universität Chemnitz, D-09107, Chemnitz, Germany

    Nico Düvelmeyer & Walter Wenzel

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  1. Nico Düvelmeyer
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  2. Walter Wenzel
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Correspondence to Nico Düvelmeyer.

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Düvelmeyer, N., Wenzel, W. A Characterization of Ordered Sets and Lattices via Betweenness Relations. Results. Math. 46, 237–250 (2004). https://doi.org/10.1007/BF03322885

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