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Fixed Point Property for Finite Ordered Sets that Contain No Crowns with 6 or More Elements

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Abstract

We prove that, for a finite ordered set P that contains no crowns with 6 or more elements, it can be determined in polynomial time if P has the fixed point property. This result is obtained by proving that every such ordered set must contain a point of rank 1 that has a unique lower cover or a retractable minimal element.

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References

  1. Donalies, M., Schröder, B.: Performance guarantees and applications for Xia’s algorithm. Discret. Math. 213, 67–86 (2000) (Proceedings of the Banach Center Minisemester on Discrete Mathematics, week on Ordered Sets)

    Article  MathSciNet  Google Scholar 

  2. Duffus, D., Goddard, T.: The complexity of the fixed point property. Order 13, 209–218 (1996)

    Article  MathSciNet  Google Scholar 

  3. Felsner, S.: 3-interval irreducible partially ordered sets. Order 11, 97–125 (1994)

    Article  MathSciNet  Google Scholar 

  4. Fofanova, T., Rutkowski, A.: The fixed point property in ordered sets of width two. Order 4, 101–106 (1987)

    Article  MathSciNet  Google Scholar 

  5. Fofanova, T., Rival, I., Rutkowski, A.: Dimension 2, fixed points and dismantlable ordered sets. Order 13, 245–253 (1994)

    MathSciNet  MATH  Google Scholar 

  6. Niederle, J.: Forbidden retracts for finite ordered sets of width at most four. Discrete Math. 308, 1774–1784 (2008)

    Article  MathSciNet  Google Scholar 

  7. Rival, I.: A fixed point theorem for finite partially ordered sets. J. Comb. Theory (A) 21, 309–318 (1976)

    Article  MathSciNet  Google Scholar 

  8. Schröder, B.: Fixed point property for 11-element sets. Order 10, 329–347 (1993)

    Article  MathSciNet  Google Scholar 

  9. Schröder, B.: Ordered Sets—an Introduction with Connections from Combinatorics to Topology, 2nd edn. Boston, Birkhäuser Verlag (2016)

    MATH  Google Scholar 

  10. Schröder, B.: The fixed point property for ordered sets of interval dimension 2. Order 34, 307–322 (2016)

    Article  MathSciNet  Google Scholar 

  11. Trotter, W. T.: Combinatorics and Partially Ordered Sets: Dimension Theory. Johns Hopkins University Press, Baltimore (1992)

    MATH  Google Scholar 

  12. Zaguia, I.: On the fixed point property for (3 + 1)-free ordered sets. Order 28, 465–479 (2011)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The author thanks the referees for constructive suggestions that improved and streamlined the presentation.

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Authors and Affiliations

  1. School of Mathematics and Natural Sciences, The University of Southern Mississippi, 118 College Avenue, #5043, Hattiesburg, MS, 39406, USA

    Bernd S. W. Schröder

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  1. Bernd S. W. Schröder
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Correspondence to Bernd S. W. Schröder.

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Schröder, B.S.W. Fixed Point Property for Finite Ordered Sets that Contain No Crowns with 6 or More Elements. Order 37, 173–178 (2020). https://doi.org/10.1007/s11083-019-09498-z

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  • DOI: https://doi.org/10.1007/s11083-019-09498-z

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