Skip to main content
SpringerLink
Log in

Pointwise Debreu Lexicographic Powers

  • Published:
Order Aims and scope Submit manuscript

Abstract

A linear ordering is Debreu (respectively, pointwise Debreu) if each of its suborderings can be mapped into it with an order-preserving function that is both injective (respectively, locally injective) and continuous (respectively, locally continuous) with respect to the order topology on both spaces. Each Debreu linear ordering is pointwise Debreu, but the converse does not hold. In the context of utility representations in mathematical economics, it has been proved that any lexicographic power with an uncountable exponent fails to be Debreu. We sharpen this result by analyzing lexicographic powers that are pointwise Debreu.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Beardon, A.F.: Analysis and Topology in Mathematical Ecomomics. University of Cambridge (1993)

  2. Beardon, A.F.: Utility theory and continuous monotonic functions. Econ. Theory 4, 531–538 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  3. Beardon, A.F., Candeal, J.C., Herden, G., Indurain, E., Mehta, G.B.: The non-existence of a utility function and the structure of non-representable preference relations. J. Math. Econ. 37, 17–38 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bosi, G., Campión, M.J., Candeal, J.C., Induráin, E.: Interval-valued representability of qualitative data: the continuous case. Int. J. Uncertain. Fuzziness Knowl.-Based Syst. 15, 299–319 (2007)

    Article  MATH  Google Scholar 

  5. Bosi, G., Candeal, J.C., Induráin, E.: Continuous representability of interval orders and biorders. J. Math. Psychol. 51, 122–125 (2007)

    Article  MATH  Google Scholar 

  6. Campión, M.J., Candeal, J.C., Induráin, E.: The existence of utility functions for weakly continuous preferences on a Banach space. Math. Soc. Sci. 51, 227–237 (2006)

    Article  MATH  Google Scholar 

  7. Campión, M.J., Candeal, J.C., Induráin, E.: Representations of binary relations through fuzzy numbers. Fuzzy Sets Syst. 157/1, 1–19 (2006)

    Article  Google Scholar 

  8. Campión, M.J., Candeal, J.C., Induráin, E., Mehta, G.B.: Order embeddings with irrational codoimain: Debreu properties of real subsets. Order 23, 343–357 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  9. Candeal, J.C., De Miguel, J.R., Induráin, E.: On a theorem of Cooper. J. Math. Anal. Appl. 258, 701–710 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  10. Candeal, J.C., Induráin, E.: Lexicographic behavior of chains. Arch. Math. 72, 145–152 (1999)

    Article  MATH  Google Scholar 

  11. Caserta, A., Giarlotta, A., Watson, S.: Debreu-like properties of utility representations. J. Math. Econ. 44, 1161–1179 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  12. Chipman, J.S.: The foundations of utility. Econometrica 28, 193–224 (1960)

    Article  MathSciNet  Google Scholar 

  13. Chipman, J.S.: On the lexicographic representations of preference orderings. In: Chipman, J.S., Hurwicz, L., Richter, M., Sonnenschein, H.F. (Eds.) Preference, Utility and Demand, pp. 276–288. Harcourt Brace and Jovanovich, New York (1971)

    Google Scholar 

  14. Cooper, J.L.B.: The foundations of thermodynamics. J. Math. Anal. Appl. 17, 172–193 (1967)

    Article  MATH  MathSciNet  Google Scholar 

  15. Debreu, G.: Representation of a preference ordering by a numerical function. In: Thrall, R., Coombs, C., Davies, R. (Eds.) Decision Processes. Wiley, New York (1954)

    Google Scholar 

  16. Engelking, R.: General Topology. Heldermann Verlag, Berlin (1989)

    MATH  Google Scholar 

  17. Giarlotta, A.: Lexicographic Products of Linear Orderings. Ph.D. Dissertation, University of Illinois at Urbana-Champaign, Urbana (2004)

  18. Giarlotta, A.: Representable lexicographic products. Order 21, 29–41 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  19. Giarlotta, A.: The representability number of a chain. Topol. its Appl. 150, 157–177 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  20. Herden, G., Mehta, G.B.: The Debreu Gap Lemma and some generalizations. J. Math. Econ. 40, 747–769 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  21. Knoblauch, V.: Lexicographic orders and preference representation. J. Math. Econ. 34, 255–267 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  22. Kuhlmann, S.: Isomorphisms of lexicographic powers of the reals. Proc. Am. Math. Soc. 123/9, 2657–2662 (1995)

    Article  MathSciNet  Google Scholar 

  23. Kunen, K.: Set Theory. An Introduction to Independence Proofs. North-Holland, Amsterdam (1980)

    MATH  Google Scholar 

  24. Rosenstein, J.G.: Linear Orderings. Academic Press, New York (1982)

    MATH  Google Scholar 

  25. Wakker, P.: Continuity of preference relations for separable topologies. Int. Econ. Rev. 29, 105–110 (1988)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Economics and Quantitative Methods, University of Catania, Catania, 95129, Italy

    Alfio Giarlotta

  2. Department of Mathematics and Statistics, York University, Toronto, M3J 1P3, Canada

    Stephen Watson

Authors
  1. Alfio Giarlotta
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Stephen Watson
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alfio Giarlotta.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Giarlotta, A., Watson, S. Pointwise Debreu Lexicographic Powers. Order 26, 377–408 (2009). https://doi.org/10.1007/s11083-009-9132-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11083-009-9132-x

Keywords

Mathematics Subject Classifications (2000)

Search

Navigation