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

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

Purchase

Purchase Print Copy

 FormatList Price Price
Add to Cart Paperback49 pages $23.00 $18.40 20% Web Discount

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)

This report is part of the RAND Corporation Paper series. The paper was a product of the RAND Corporation from 1948 to 2003 that captured speeches, memorials, and derivative research, usually prepared on authors' own time and meant to be the scholarly or scientific contribution of individual authors to their professional fields. Papers were less formal than reports and did not require rigorous peer review.

Our mission to help improve policy and decisionmaking through research and analysis is enabled through our core values of quality and objectivity and our unwavering commitment to the highest level of integrity and ethical behavior. To help ensure our research and analysis are rigorous, objective, and nonpartisan, we subject our research publications to a robust and exacting quality-assurance process; avoid both the appearance and reality of financial and other conflicts of interest through staff training, project screening, and a policy of mandatory disclosure; and pursue transparency in our research engagements through our commitment to the open publication of our research findings and recommendations, disclosure of the source of funding of published research, and policies to ensure intellectual independence. For more information, visit www.rand.org/about/research-integrity.

The RAND Corporation is a nonprofit institution that helps improve policy and decisionmaking through research and analysis. RAND's publications do not necessarily reflect the opinions of its research clients and sponsors.