Ideals in ordered sets, a unifying approach

  • Josef NiederleEmail author


A common generalization of different notions of ideals met in literature together with the associated notions of distributivity in ordered sets are studied. A restricted prime ideal theorem is proved. Moreover, a new characterization of algebraic topped intersection structures is presented.


Algebraic topped intersection structure ideal pseudoideal Frink ideal Doyle pseudoideal S-ideal primeS-ideal S-distributive ordered set primeS-ideal theorem 

Mathematics Subject Classification (2000)

06A06 06A15 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Abian, A., Amin, W. A.,Existence of prime ideals and ultrafilters in partially ordered sets, Czech. Math. J.,40 1990, 159–163.MathSciNetGoogle Scholar
  2. [2]
    Davey, B. A., Priestley, H. A.,Introduction to Lattices and Order, Cambridge University Press 1990.Google Scholar
  3. [3]
    Doyle, W.,An arithmetical theorem for partially ordered sets, Bull. Amer. Math. Soc.,56 1950, 366.Google Scholar
  4. [4]
    Frink, O.,Ideals in partially ordered sets, Amer. Math. Monthly,61 1954, 223–234.zbMATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    Halaš, R., Rachůnek, J.,Polars and prime ideals in ordered sets, Discuss. Math., Algebra Stoch. Methods,15 1995, 43–59.Google Scholar
  6. [6]
    Niederle, J.,Boolean and distributive ordered sets: characterization and representation by sets, Order,12 1995, 189–210.zbMATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    Venkatanarasimhan, P. V.,Pseudo-complements in posets, Proc. Amer. Math. Soc.,28 1971, 9–17.zbMATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    Wright, J. B., Wagner, E. G., Thatcher, J. W.,A uniform approach to inductive posets and inductive closure, Theoret. Comput. Sci., 7 1978, 57–77.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer 2006

Authors and Affiliations

  1. 1.Masarykova universita, Přírodovědecká fakultaKatedra algebry a geometrieBrnoCzechoslovakia

Personalised recommendations