Skip to main content
SpringerLink
Log in

On Minimal Prime Graphs and Posets

  • Published:
Order Aims and scope Submit manuscript

Abstract

We show that there are four infinite prime graphs such that every infinite prime graph with no infinite clique embeds one of these graphs. We derive a similar result for infinite prime posets with no infinite chain or no infinite antichain.

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. Bondy, J.A., Murty, U.S.R.: Graph Theory, Basic Graph Theory, Graduate Texts in Mathematics, vol. 244, 651 pp. Springer, New York (2008)

    Google Scholar 

  2. Bouchet, A.: Étude combinatoire des ordonnés finis. Thèse de Doctorat d’État, Université Scientifique et Médicale de Grenoble (1971)

  3. Bouchet, A.: Codages et dimensions de relations binaires. In: Orders: Description and Roles (L’Arbresle, 1982), vol. 99, pp. 387–396. North-Holl. Math. Stud. North-Holland, Amsterdam (1984)

    Google Scholar 

  4. Boudabbous, Y., Pouzet, M.: The morphology of infinite tournaments. Application to the growth of their profile. Presented at the CGCS-2007 Conference in Honor of Michel Deza. European Journal of Combinatorics (2009). doi:10.1016/j.ejc2009.03.027

    Google Scholar 

  5. Duffus, D., Pouzet, M., Rival, I.: Complete ordered sets with no infinite antichains. Discrete Math. 35, 39–52 (1981)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  7. Fraïssé, R.: L’intervalle en théorie des relations, ses généralisations, filtre intervallaires et clôture d’une relation, in Orders: description and roles (L’Arbresle, 1982), vol. 99, pp. 343–358. North-Holl. Math. Stud. North-Holland, Amsterdam (1984)

  8. Gallai, T.: Transitiv orientierbare Graphen. Acta Math. Acad. Sci. Hung. 18, 25–66 (1967)

    Article  MATH  MathSciNet  Google Scholar 

  9. Ille, P.: A characterization of the indecomposable and infinite graphs. Glob. J. Pure Appl. Math. 3, 272–285 (2005)

    MathSciNet  Google Scholar 

  10. Kelly, D.: Comparability graphs. In: Rival, I. (ed.) Graphs and Orders. NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 147, pp. 3–40 (1985)

  11. Pouzet, M., Sikaddour, H., Zaguia, N.: Which posets have a scattered MacNeille completion? Algebra Univers. 53, 287–299 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  12. Ramsey, F.P.: On a problem of formal logic. Proc. Lond. Math. Soc. II 30, 264–286 (1930)

    Article  Google Scholar 

  13. Riguet, J.: Sur les ensembles réguliers de relations binaires, les relations de Ferrers, vol. 231, pp. 936–937. CRAS, Paris (1950)

    Google Scholar 

  14. Sumner, D.P.: Point determination in graphs. Discrete Math. 5, 179–187 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  15. Sumner, D.P.: Graphs indecomposable with respect to the X-join. Discrete Math. 6, 281–298 (1973)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. ICJ, Université Claude-Bernard Lyon 1, 43 Bd. 11 Novembre 1918, 69622, Villeurbanne cedex, France

    Maurice Pouzet

  2. Department of Mathematics and Statistics, The University of Calgary, 2500 University Drive N. W., Calgary, Alberta, Canada, T2N 1N4

    Maurice Pouzet

  3. Department of Mathematics & Statistics, Sultan Qaboos University, P.O. Box 36, Al-Khoud 123, Muscat, Sultanate of Oman

    Imed Zaguia

Authors
  1. Maurice Pouzet
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Imed Zaguia
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Imed Zaguia.

Additional information

This research, under the auspices of the French-Tunisian CMCU “utils mathématiques pour l’Informatique” 05S1505, was done while the second author was visiting Sultan Qaboos University. The support provided by the university is gratefully acknowledged.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Pouzet, M., Zaguia, I. On Minimal Prime Graphs and Posets. Order 26, 357–375 (2009). https://doi.org/10.1007/s11083-009-9131-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11083-009-9131-y

Keywords

AMS Subject Classifications (2000)

Search

Navigation