Partial Orders of Dimension 2, Interval Orders, and Interval Graphs.

by K. A. Baker, Peter C. Fishburn, Fred S. Roberts

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A contribution to the theory of measurement, relevant to the measurement of preferences and the generalization of one-dimensional utility functions. The dimension of a partial order is defined as the cardinality of the smallest collection of linear orders whose intersection gives that partial order. It is shown that the dimension of a partial order is at most 2 if and only if its incomparability graph is a comparability graph. Early work of Dushnik and Miller is combined with more recent results to provide a new characterization (axiomatization) of partial orders with dimensions at most 2, and these are related to lattices with planar Hasse diagrams. The class of partial orders with dimensions at most 2 is shown to be not finitely axiomatizable. The relationship between dimensionality and other types of binary relations is illustrated-- among them, weak orders, interval orders, semiorders, interval graphs, and the breadth of a partial order. 49 pp. Ref. (MW)

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