Numerical representability of semiorders

In the framework of the analysis of orderings whose associated indifference relation is not necessarily transitive, we study the structure of a semiorder, and its representability through a real-valued function and a threshold. Inspired in a recent characterization of the representability of interval orders, we obtain a full characterization of the existence of numerical representations for semiorders. This is an extension to the general case of the classical Scott-Suppes theorem concerning the representability of semiorders defined on finite sets. © 2002 Elsevier Science B.V. All rights reserved.

[1]  S. Gensemer.  On relationships between numerical representations of interval orders and semiorders , 1987 .

[2]  P. Fishburn.  Intransitive indifference with unequal indifference intervals , 1970 .

[3]  S. Gensemer.  Continuous semiorder representations , 1987 .

[4]  Patrick Suppes,et al.  Basic measurement theory , 1962 .

[5]  Eugene Galanter,et al.  Handbook of mathematical psychology: I. , 1963 .

[6]  Juan Carlos Candeal,et al.  Lexicographic behaviour of chains , 1999 .

[7]  S. Gensemer.  On numerical representations of semiorders , 1988 .

[8]  M. Pirlot.  Minimal representation of a semiorder , 1990 .

[9]  E. Induráin,et al.  Representability of Interval Orders , 1998 .

[10]  D. Jamison,et al.  The Nature of Equilibrium with Semiordered Preferences , 1977 .

[11]  Jutta Mitas.  Interval orders based on arbitrary ordered sets , 1995, Discret. Math..

[12]  P. Fishburn.  Semiorders and risky choices , 1968 .

[13]  Philippe Vincke,et al.  Semiorders - Properties, Representations, Applications , 1997, Theory and decision library: series B.

[14]  Jean-Paul Doignon,et al.  On realizable biorders and the biorder dimension of a relation , 1984 .

[15]  G. Bosi,et al.  Representing preferences with nontransitive indifference by a single real-valued function☆ , 1995 .

[16]  R. Luce.  Semiorders and a Theory of Utility Discrimination , 1956 .

[17]  Y. Ng.  Bentham or Bergson? Finite Sensibility, Utility Functions and Social Welfare Functions , 1975 .

[18]  Esteban Induráin Eraso.  Preferencias con indiferencia no transitiva: problemas abiertos relativos a órdenes-intervalo , 2002 .

[19]  Marc Pirlot,et al.  Synthetic description of a semiorder , 1991, Discret. Appl. Math..

[20]  A. Tversky.  Intransitivity of preferences. , 1969 .

[21]  R. M. Adelson,et al.  Utility Theory for Decision Making , 1971 .

[22]  P. Fishburn.  Interval representations for interval orders and semiorders , 1973 .

[23]  W. E. Armstrong.  A NOTE ON TUE THEORY OF CONSUMER'S BEHAVIOUR , 1950 .

[24]  D. Bridges.  Representing interval orders by a single real-valued function , 1985 .

[25]  D. Scott.  Measurement structures and linear inequalities , 1964 .

[26]  R. Agaev,et al.  Interval choice: classic and general cases , 1993 .

[27]  P. Vincke.  LINEAR UTILITY FUNCTIONS ON SEMIORDERED MIXTURE SPACES , 1980 .

[28]  Leonard Gillman,et al.  Rings of continuous functions , 1961 .

[29]  J. Aczel,et al.  A Short Course on Functional Equations: Based Upon Recent Applications to the Social and Behavioral Sciences , 1986 .

[30]  D. Krantz.  Extensive Measurement in Semiorders , 1967, Philosophy of Science.

[31]  Kenneth L. Manders.  On jnd representations of semiorders , 1981 .

[32]  Patrick Suppes,et al.  Foundational aspects of theories of measurement , 1958, Journal of Symbolic Logic.

[33]  Peter C. Fishburn,et al.  Intransitive Indifference in Preference Theory: A Survey , 1970, Oper. Res..

[34]  Esteban Induráin Eraso,et al.  APLICACIONES BIVARIANTES QUE REPRESENTAN SEMIGRUPOS ORDENADOS , 1996 .