, Volume 12, Issue 2, pp 189–210 | Cite as

Boolean and distributive ordered sets: Characterization and representation by sets

  • Josef Niederle


Boolean ordered sets generalize Boolean lattices, and distributive ordered sets generalize distributive lattices. Ideals, prime ideals, and maximal ideals in ordered sets are defined, and some well-known theorems on Boolean lattices, such as the Glivenko-Stone theorem and the Stone representation theorem, are generalized to Boolean ordered sets. A prime ideal theorem for distributive ordered sets is formulated, and the Birkhoff representation theorem is generalized to distributive ordered sets. Fundamental are the embedding theorems for Boolean ordered sets and for distributive ordered sets.

Mathematics Subject Classification (1991)


Key words

Boolean ordered set distributive ordered set ideal prime ideal maximal ideal representation by sets 


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  1. 1.
    Chajda, Ivan (1992) Complemented ordered sets,Archivum Math. (Brno)28, 25–34.Google Scholar
  2. 2.
    Chajda, Ivan and Rachůnek, Jiří (1989) Forbidden configurations for distributive and modular ordered sets,Order 5, 407–423.Google Scholar
  3. 3.
    Frink, Orrin (1954) Ideals in partially ordered sets,Amer. Math. Monthly 61, 223–234.Google Scholar
  4. 4.
    Halaš, Radomír (1993) Pseudocomplemented ordered sets,Archivum Math. (Brno)29, 153–160.Google Scholar
  5. 5.
    Larmerová, Jana and Rachůnek, Jiří (1988) Translations of distributive and modular ordered sets,Acta Univ. Palack. Olom., Math. 27, 13–23.Google Scholar
  6. 6.
    Niederle, Josef (1991) Natural generalization of some lattice theory concepts to partially ordered sets,Czechoslovak Math. J. 41, 297–299.Google Scholar
  7. 7.
    Paseka, Jan (in preparation) Note on distributive posets.Google Scholar

Copyright information

© Kluwer Academic Publishers 1995

Authors and Affiliations

  • Josef Niederle
    • 1
  1. 1.Katedra algebry a geometrie přirodovědecké fakulty Masarykovy universityBrnoCzechia

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