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

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

Published 1970

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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