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Unified Representability of Total Preorders and Interval Orders through a Single Function: The Lattice Approach

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Abstract

We introduce a new approach that deals, jointly and in a unified manner, with the topics of numerical (continuous) representability of total preorders and interval orders. This setting is based on the consideration of increasing scales and the systematic use of a particular kind of codomain, that has a key lattice theoretical structure of a completely distributive lattice and allows us to use a single function (taking values in that codomain) in order to represent both kinds of binary relations.

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References

  1. Abrísqueta, F.J., Candeal, J.C., Induráin, E., Zudaire, M.: Scott-suppes representability of semiorders: internal conditions. Math. Soc. Sci. 57, 245–261 (2009)

    Article  MATH  Google Scholar 

  2. Alcantud, J.C.R., Bosi, G., Campión, M.J., Candeal, J.C., Induráin, E., Rodríguez-Palmero, C.: Continuous utility functions through scales. Theory Decis. 64, 479–494 (2008)

    Article  MATH  Google Scholar 

  3. Atanassov, K.T.: Intuitionistic fuzzy sets. Fuzzy Sets Syst. 20, 87–96 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bellman, R., Kalaba, R., Zadeh, L.A.: Abstraction and pattern classification. J. Math. Anal. Appl. 13, 1–7 (1966)

    Article  MATH  MathSciNet  Google Scholar 

  5. Birkhoff, G.: Lattice Theory, 3rd edn. American Mathematical Society, Providence (1967)

    MATH  Google Scholar 

  6. Bosi, G.: Continuous representation of interval orders by means of decreasing scales. Mat. Vesn. 58, 111–117 (2006)

    MATH  MathSciNet  Google Scholar 

  7. 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(3), 299–319 (2007)

    Article  MATH  Google Scholar 

  8. Bosi, G., Campión, M.J., Candeal, J.C., Induráin, E., Zuanon, M.E.: Numerical isotonies of preordered semigroups through the concept of a scale. Math. Pannon. 16(1), 65–77 (2005)

    MATH  MathSciNet  Google Scholar 

  9. Bosi, G., Campión, M.J., Candeal, J.C., Induráin, E., Zuanon, M.E.: Isotonies on ordered cones through the concept of a decreasing scale. Math. Soc. Sci. 54, 115–127 (2007)

    Article  MATH  Google Scholar 

  10. Bosi, G., Candeal, J.C., Induráin, E., Olóriz, E., Zudaire, M.: Numerical representations of interval orders. Order 18, 171–190 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  11. Bosi, G., Herden, G.: On the structure of completely useful topologies. Appl. Gen. Topol. 3(2), 145–167 (2002)

    MATH  MathSciNet  Google Scholar 

  12. Bosi, G., Isler, R.: A full characterization of continuous representability of interval orders. Math. Pannon. 18(1), 125–134 (2007)

    MATH  MathSciNet  Google Scholar 

  13. Bridges, D.S., Mehta, G.B.: Representations of Preference Orderings. Springer, Berlin (1995)

    MATH  Google Scholar 

  14. Burgess, D.C.J., Fitzpatrick, M.: On separation axioms for certain types of ordered topological space. Math. Proc. Camb. Philos. Soc. 82, 59–65 (1977)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

  16. Campión, M.J., Candeal, J.C., Induráin, E., Zudaire, M.: Continuous representability of semiorders. J. Math. Psychol. 52, 48–54 (2008)

    Article  MATH  Google Scholar 

  17. Candeal, J.C., Gutiérrez García, J., Induráin, E.: Universal codomains to represent interval orders. Int. J. Uncertain. Fuzziness Knowl.-Based Syst. 17(2), 197–219 (2009)

    Article  MATH  Google Scholar 

  18. Candeal, J.C., Induráin, E., Zudaire, M.: Numerical representability of semiorders. Math. Soc. Sci. 43(1), 61–77 (2002)

    Article  MATH  Google Scholar 

  19. Doignon, J.P., Ducamp, A., Falmagne, J.C.: On realizable biorders and the biorder dimension of a relation. J. Math. Psychol. 28, 73–109 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  20. Felsner, S., Gustedt, J., Morvan, M.: Interval reductions and extensions of orders: bijections to chains in lattices. Order 15, 221–246 (1999)

    Article  MathSciNet  Google Scholar 

  21. Fishburn, P.C.: Interval representations for interval orders and semiorders. J. Math. Psychol. 10, 91–105 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  22. Fishburn, P.C.: Interval Orders and Interval Graphs. Wiley, New York (1985)

    MATH  Google Scholar 

  23. Fishburn, P.C., Monjardet, B.: Norbert Wiener on the theory of measurement (1914, 1915, 1921). J. Math. Psychol. 36(2), 165–184 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  24. Gierz, G., Hofmann, K.H., Keimel, K., Lawson, J.D., Mislove, M., Scott, D.S.: Continuous Lattices and Domains. Cambridge University Press, Cambridge (2003)

    MATH  Google Scholar 

  25. Gutiérrez García, J., Kubiak, T., de Prada Vicente, M.A.: Insertion of lattice-valued and hedgehog-valued functions. Topol. Appl. 153, 1458–1475 (2006)

    Article  MATH  Google Scholar 

  26. Gutiérrez García, J., Kubiak, T., de Prada Vicente, M.A.: Generating and inserting continuous functions with values in bounded complete domains and hedgehog-like structures. Houst. J. Math. 34, 123–144 (2008)

    Google Scholar 

  27. Gutiérrez García, J., Kubiak, T., de Prada Vicente, M.A.: Controlling disjointness with a hedgehog. Houst. J. Math. 35, 469–484 (2009)

    Google Scholar 

  28. Gutiérrez García, J., Rodabaugh, S.E.: Order-theoretic, topological, categorical redundancies of interval-valued sets, grey sets, vague sets, interval-valued “intuitionistic” sets, “intuitionistic” fuzzy sets and topologies. Fuzzy Sets Syst. 156, 445–484 (2005)

    Article  MATH  Google Scholar 

  29. Herden, G.: Topological spaces for which every continuous total preorder can be represented by a continuous utility function. Math. Soc. Sci. 22, 123–136 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  30. Herden, G., Pallack, A.: Useful topologies and separable systems. Appl. Gen. Topol. 1(1), 61–82 (2000)

    MATH  MathSciNet  Google Scholar 

  31. Höhle, U., Kubiak, T.: Approximating orders in meet-continuous lattices and regularity axioms in many valued topology. Order 25, 9–17 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  32. Luce, R.D.: Semiorders and a theory of utility discrimination. Econometrica 24, 178–191 (1956)

    Article  MATH  MathSciNet  Google Scholar 

  33. Marichal, J.L., Mesiar, R.: Aggregation on finite ordinal scales by scale independent functions. Order 21, 155–180 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  34. Olóriz, E., Candeal, J.C., Induráin, E.: Representability of interval orders. J. Econ. Theory 78(1), 219–227 (1998)

    Article  MATH  Google Scholar 

  35. Raney, G.N.: A subdirect-union representation for completely distributive complete lattices. Proc. Am. Math. Soc. 4, 518–522 (1953)

    Article  MATH  MathSciNet  Google Scholar 

  36. Scott, D., Suppes, P.: Foundational aspects of theories of measurement. J. Symb. Log. 23, 113–128 (1958)

    Article  MathSciNet  Google Scholar 

  37. Zadeh, L.A.: Fuzzy sets. Inf. Control 8, 338–353 (1965)

    Article  MATH  MathSciNet  Google Scholar 

  38. Zhang, D.-X.: Meet continuous lattices, limit spaces, and L-topological spaces. Fuzzy Sets Syst. 140, 479–487 (2003)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Dipartimento di Matematica Applicata “Bruno de Finetti”, Universitá di Trieste, Piazzale Europa 1, 34127, Trieste, Italy

    Gianni Bosi

  2. Departamento de Matemáticas, Universidad del País Vasco-Euskal Herriko Unibertsitatea, Apdo. 644, 48080, Bilbao, Spain

    Javier Gutiérrez García

  3. Departamento de Matemáticas, Universidad Pública de Navarra, Campus Arrosadía, 31006, Pamplona, Spain

    Esteban Induráin

Authors
  1. Gianni Bosi
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  2. Javier Gutiérrez García
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  3. Esteban Induráin
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Correspondence to Javier Gutiérrez García.

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Bosi, G., Gutiérrez García, J. & Induráin, E. Unified Representability of Total Preorders and Interval Orders through a Single Function: The Lattice Approach. Order 26, 255–275 (2009). https://doi.org/10.1007/s11083-009-9124-x

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  • DOI: https://doi.org/10.1007/s11083-009-9124-x

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