Skip to main content
Springer Nature Link
Log in

On a Possible Continuous Analogue of the Szpilrajn Theorem and its Strengthening by Dushnik and Miller

  • Published:
Order Aims and scope Submit manuscript

Abstract

The Szpilrajn theorem and its strengthening by Dushnik and Miller belong to the most quoted theorems in many fields of pure and applied mathematics as, for instance, order theory, mathematical logic, computer sciences, mathematical social sciences, mathematical economics, computability theory and fuzzy mathematics. The Szpilrajn theorem states that every partial order can be refined or extended to a total (linear) order. The theorem by Dushnik and Miller states, moreover, that every partial order is the intersection of its total (linear) refinements or extensions. Since in mathematical social sciences or, more general, in any theory that combines the concepts of topology and order one is mainly interested in continuous total orders or preorders in this paper some aspects of a possible continuous analogue of the Szpilrajn theorem and its strengthening by Dushnik and Miller will be discussed. In particular, necessary and sufficient conditions for a topological space to satisfy a possible continuous analogue of the Dushnik-Miller theorem will be presented. In addition, it will be proved that a continuous analogue of the Szpilrajn theorem does not hold in general. Further, necessary and in some cases necessary and sufficient conditions for a topological space to satisfy a possible continuous analogue of the Szpilrajn theorem will be presented.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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. Abian, A.: Linear extensions of partial orders. Nordisk Mat. Tidskr. 22, 21–22 (1974)

    MATH  MathSciNet  Google Scholar 

  2. Blanchard, N.: Cardinal and ordinal theories about fuzzy sets, fuzzy information and decision processes, pp. 149–157. North-Holland, Amsterdam (1982)

    Google Scholar 

  3. Blanchard, N.: Embedding a fuzzy ordering into a fuzzy linear fuzzy ordering (Szpilrajn-Marczewski-like theorems), Fuzzy information, knowledge representation and decision analysis (Marseille 1983). IFAC Proc. Ser. 6, 123–127 (1984)(IFAC, Laxenburg)

    MathSciNet  Google Scholar 

  4. Bondesen, A.: Extension of circle-free relations. Nordisk Mat. Tidskr. 25–26, 155–156, 209 (1978)

    MathSciNet  Google Scholar 

  5. Bosi, G., Herden, G.: On a strong continuous analogue of the Szpilrajn theorem and its strengthening by Dushnik and Miller. Order 22, 329–342 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  6. Bossert, W.: Intersections of quasi-orderings: An alternative proof. Order 16, 221–225 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  7. Bossert, W., Sprumont, Y., Suzumura, K.: Upper semicontinuous extensions of binary relations. J. Math. Econ. 37, 231–246 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  8. Bridges, D.S., Mehta, G.B.: Representation of preference orderings. Lecture Notes in Economics and Mathematical Systems 422. Springer, Berlin Heidelberg New York (1995)

    Google Scholar 

  9. Candeal, J.C., Hervés, C., Induráin, E.: Some results on representation and extension of preferences. J. Math. Econ. 29, 75–81 (1998)

    Article  MATH  Google Scholar 

  10. Chipman, J.: The foundation of utility. Econometrica. 28, 193–224 (1960)

    Article  MathSciNet  Google Scholar 

  11. Cigler, J., Reichel, H.C.: Topologie, Bibliographisches Institut. Mannheim, Germany (1978)

  12. Debreu, G.: Representation of preference orderings by a numerical function. In: Thrall, R., Coombs, C.C., Davis, R. (eds.) Decision Processes, pp. 159–166. Wiley, New York (1954)

    Google Scholar 

  13. Debreu, G.: Continuity properties of Paretian utility. Int. Econ. Rev. 5, 285–293 (1964)

    Article  Google Scholar 

  14. Diaye, M.A.: Variable intervals model. Math. Soc. Sci. 38, 21–33 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  15. Donaldson, D., Weymark, J.A.: A quasiordering is the intersection of orderings. J. Econ. Theory 78, 382–387 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  16. Downey, R.G.: Computability theory and linear orderings. Handbook of Recursive Mathematics, vol. 2, pp. 823–976. North Holland, Amsterdam (1998)

    Google Scholar 

  17. Duggan, J.: A general extension theorem for binary relations. J. Econ. Theory 86, 1–16 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  18. Dushnik, B., Miller, E.W.: Partially ordered sets. Amer. J. Math. 63, 600–610 (1941)

    Article  MATH  MathSciNet  Google Scholar 

  19. Erné, M.: Einführung in die Ordnungstheorie, Bibliographisches Institut. Mannheim, Germany (1982)

  20. Estévez, M., Hervés, C.: On the existence of continuous preference orderings without utility representation. J. Math. Econ. 24, 305–309 (1995)

    Article  MATH  Google Scholar 

  21. Fay, H.T.: A note on cirle free relations. Nordisk Mat. Tidskr. 24, 52 (1976)

    MathSciNet  Google Scholar 

  22. Felgner, U., Truss, J.K.: The independence of the prime ideal theorem from the order-extension principle. J. Symb. Log. 64, 199–215 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  23. Fishburn, P.C.: Utility Theory for Decision Making. Wiley, New York (1970)

    MATH  Google Scholar 

  24. Fishburn, P.C.: The Theory of Social Choice. Princeton University Press, Princeton, NJ (1973)

    MATH  Google Scholar 

  25. Fishburn, P.C.: Linear extensions of additive partial orders. Order 14, 153–169 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  26. Gierz, G., Hofmann, K.H., Keimel, K., Lawson, J.D., Mislove, M.W., Scott, D.S.: A Compendium of Continuous Lattices. Springer, Berlin Heidelberg New York (1980)

    MATH  Google Scholar 

  27. Gierz, G., Hofmann, K.H., Keimel, K., Lawson, J.D., Mislove, M.W., Scott, D.S.: Continuous Lattices and Domains, Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, UK (2003)

    Google Scholar 

  28. Hajnal, A., Hamburger, P.: Set Theory. Cambridge University Press, Cambridge, UK (1999)

    MATH  Google Scholar 

  29. Hashimoto, H.: Szpilrajn’s theorem on fuzzy orderings. Fuzzy Sets Syst. 10, 101–108 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  30. Herden, G., Mehta, G.B.: Some aspects of a semicontinuous analogue of the Szpilrajn theorem. Universität Duisburg-Essen, Campus Essen, preprint (2005)

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

    MATH  MathSciNet  Google Scholar 

  32. Herden, G., Pallack, A.: On the continuous analogue of the Szpilrajn theorem I. Math. Soc. Sci. 43, 115–134 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  33. Höhle, U., Blanchard, N.: Partial ordering in L-underdeterminate sets. Inform. Sc. 35, 133–144 (1985)

    Article  MATH  Google Scholar 

  34. Jaffray, J.-Y.: Semicontinuous extension of a partial order. J. Math. Econ. 2, 395–406 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  35. Janicki, R., Koutny, M.: Fundamentals of modelling concurrency using discrete relational structures. Acta Inform. 34, 367–388 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  36. Kontolatou, A., Stabakis, J.: Embedding groups into linear or lattice structures. Bull. Calcutta Math. Soc. 82, 290–297 (1990)

    MATH  MathSciNet  Google Scholar 

  37. Lahiri, S.: A simple proof of Suzumura’s extension theorem for finite domains with applications. J. Appl. Math. Decis. Sci. 6, 183–190 (2002)

    MATH  MathSciNet  Google Scholar 

  38. Levin, V.: Measurable utility theorems for closed and lexicographic preference relations. Soviet Math. Dokl. 27, 639–643 (1983)

    MATH  Google Scholar 

  39. Martínez-Legaz, J-E., Singer, I.: Compatible preorders and linear operators on \( \mathbb{R}^{ \ltimes } \). Linear Algebra Appl. 153, 53–66 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  40. Morgado, J.: Note on quasi-orders, partial orders and orders. Portugal. Math. 32, 1–7 (1973)

    MATH  MathSciNet  Google Scholar 

  41. Moulin, H.: Choice functions over a finite set: a summary. Soc. Choice Welf. 2, 147–160 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  42. Nachbin, L.: Topology and Order. Van Nostrand, Princeton (1965)

    MATH  Google Scholar 

  43. Nehring, K., Puppe, C.: Extended partial orders: a unifying structure for abstract choice theory. Ann. Oper. Res. 80, 27–48 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  44. Ovchinnikov, S.V.: Representations of transitive fuzzy relations, Aspects of vagueness (Las Palmas, 1982). Theory Decis. Libr., Reidel, 105–118 (1984)(Dordrecht)

  45. Roubens, M., Vincke, P.: Preference modelling. Lectures Notes in Economics and Mathematical Systems, 250. Springer, Berlin Heidelberg New York (1985)

    Google Scholar 

  46. Roy, D.K.: Effective extensions of partial orders. Z. Math. Log. Grundl. Math. 36, 233–236 (1990)

    Article  MATH  Google Scholar 

  47. Rozen, V.V.: Coding of ordered sets, Ordered sets and lattices, No. 10. 88–96. Saratov. Gos. Univ., Saratov (1991)

  48. Sholomov, L.A.: Explicit form of neutral decision rules for basic rationality conditions. Math. Soc. Sci. 39, 81–107 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  49. Sprumont, Y.: Paretian quasi-orders: the regular two agent case. J. Econ. Theory 101, 437–456 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  50. Steen, L.A., Seebach, J.A.: Counterexamples in Topology. Dover, New York (1995)

    Google Scholar 

  51. Stehr, M.O.: Thinking in cycles. Proceedings of the 19th International Conference on application and theory of Petrinets, pp. 205–225. Springer, Berlin Heidelberg New York June 22–26 1998

    Google Scholar 

  52. Suck, R.: A theorem on order extensions – embeddability of a system of weak orders to meet solvability constraints. J. Math. Psychol. 38, 128–134 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  53. Szpilrajn, E.: Sur l’extension de l’ordre partial. Fundam. Math. 16, 386–389 (1930)

    MATH  Google Scholar 

  54. Wang, D.B.: On the fuzzy Szpilrajn theorem. Neimenggu Daxue Xuebao 14, 67–74 (1983)

    MathSciNet  Google Scholar 

  55. Weymark, J.A.: A generalization of Moulin’s Pareto extension theorem. Math. Soc. Sci. 39, 235–240 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  56. Wu, K.P.: Countable satiabilities of preferences. Acta Math. Sinica 10, 36–43 (1994)

    MATH  MathSciNet  Google Scholar 

  57. Yi, G.: Continuous extension of preferences. J. Math. Econ. 22, 547–555 (1993)

    Article  MATH  Google Scholar 

  58. Zadeh, L.A.: Similarity relations and fuzzy orderings. Inform. Sc. 3, 177–200 (1971)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

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

    Gianni Bosi

  2. Universität Duisburg-Essen, Campus Essen, Fachbereich 6 (Mathematik) Universitätsstraße 3, 45117, Essen, Germany

    Gerhard Herden

Authors
  1. Gianni Bosi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gerhard Herden
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gerhard Herden.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bosi, G., Herden, G. On a Possible Continuous Analogue of the Szpilrajn Theorem and its Strengthening by Dushnik and Miller. Order 23, 271–296 (2006). https://doi.org/10.1007/s11083-006-9047-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11083-006-9047-8

Key words

Mathematics Subject Classifications (2000)