Skip to main content
SpringerLink
Log in

Standard completions for quasiordered sets

  • Research Article
  • Published:
Semigroup Forum Aims and scope Submit manuscript

Abstract

A standard completion for a quasiordered set Q is a closure system whose point closures are the principal ideals of Q. We characterize the following types of standard completions by means of their closure operators:

  1. (i)

    V-distributive completions,

  2. (ii)

    Completely distributive completions,

  3. (iii)

    A-completions (i.e. standard completions which are completely distributive algebraic lattices),

  4. (iv)

    Boolean completions.

Moreover, completely distributive completions are described by certain idempotent relations, and the A-completions are shown to be in one-to-one correspondence with the join-dense subsets of Q. If a pseudocomplemented meet-semilattice Q has a Boolean completion ℭ, then Q must be a Boolean lattice and ℭ its MacNeille completion.

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. R. BALBES and Ph. DWINGER: Distributive lattices. University of Missouri Press, Columbia, Mo., 1974.

    MATH  Google Scholar 

  2. V. K. BALACHANDRAN: A characterization of ΣΔ-rings of subsets. Fund. Math. 41 (1954), 38–41.

    MATH  MathSciNet  Google Scholar 

  3. B. BANASCHEWSKI: Hüllensysteme und Erweiterung von Quasi-Ordnungen. Zeitschr. für math. Logik und Grundlagen der Math. 2 (1956), 117–130.

    MATH  MathSciNet  Google Scholar 

  4. H.-J. BANDELT and M. ERNÉ: Representations and embeddings ofM-distributive lattices. Houston J. Math. (to appear).

  5. G. BIRKHOFF: Lattice theory. AMS Colloquium Publ. Vol. 25, 3d ed., Rhode Island, 1967.

  6. A. BISHOP: A universal mapping property for a lattice completion. Algebra Universalis 6 (1976), 81–84.

    Article  MATH  MathSciNet  Google Scholar 

  7. G. BRUNS: Darstellungen und Erweiterungen geordneter Mengen I/II. J. Reine Angew. Math. 209 (1962), 167–200; ibid. 210 (1962), 1–23.

    MathSciNet  MATH  Google Scholar 

  8. J. R. BÜCHI: Representations of complete lattices by sets. Portugal. Math. 11 (1952), 151–167.

    Google Scholar 

  9. P. CRAWLEY: Regular embeddings which preserve lattice structure. Proc. AMS 13 (1962), 748–752.

    Article  MATH  MathSciNet  Google Scholar 

  10. M. ERNÉ: Scott convergence and Scott topology in partially ordered sets II. In: Continuous lattices; Proceedings Bremen 1979, Lecture Notes in Math. 871, Springer-Verlag, Berlin-Heidelberg-New York, 1981.

    Google Scholar 

  11. M. ERNÉ: Distributivgesetze und Dedekind’sche Schnitte. Festschrift der Braunschweig. Wiss. Ges. zum 150. Geburtstag Richard Dedekinds, Braunschweig (to appear).

  12. O. FRINK: Ideals in partially ordered sets. Am. Math. Monthly 61 (1954), 223–234.

    Article  MATH  MathSciNet  Google Scholar 

  13. O. FRINK: Pseudo-complements in semi-lattices. Duke Math. J. 29 (1962), 505–514.

    Article  MATH  MathSciNet  Google Scholar 

  14. G. GIERZ, K. H. HOFMANN, K. KEIMEL, J. D. LAWSON, M. MISLOVE, D. S. SCOTT: A compendium of continuous lattices. Springer-Verlag, Berlin-Heidelberg-New York, 1980.

    MATH  Google Scholar 

  15. V. GLIVENKO: Sur quelques points de la logique de M. Brouwer. Bull. Acad. Sci. Belgique 15 (1929), 183–188.

    Google Scholar 

  16. G. GRÄTZER: General lattice theory. Birkhäuser Verlag, Basel-Stuttgart, 1978.

    Google Scholar 

  17. R.-E. HOFFMANN: Sobrification of partially ordered sets. Semigroup Forum 17 (1979), 123–138.

    Article  MATH  MathSciNet  Google Scholar 

  18. R.-E. HOFFMANN: Continuous posets, prime spectra of completely distributive lattices and Hausdorff compactifications. In: Continuous lattices, Proceedings Bremen 1979, Lecture Notes in Math. 871, Springer-Verlag, Berlin-Heidelberg-New York, 1981.

    Google Scholar 

  19. T. KATRIŇÁK: Pseudokomplementäre Halbverbände. Mat. casopis 18 (1968), 121–143.

    MATH  Google Scholar 

  20. O. ORE: Combinations of closure relations. Ann. of Math. 44 (1943), 514–533.

    Article  MathSciNet  Google Scholar 

  21. G. N. RANEY: Completely distributive complete lattices. Proc. AMS 3 (1952), 677–680.

    Article  MATH  MathSciNet  Google Scholar 

  22. G. N. RANEY: A subdirect-union representation for completely distributive complete lattices. Proc. AMS 4 (1953), 518–522.

    Article  MATH  MathSciNet  Google Scholar 

  23. J. SCHMIDT: Einige grundlegende Begriffe und Sätze aus der Theorie der Hüllenoperatoren. Ber. Math. Tagung Berlin (1953), 21–48.

  24. J. SCHMIDT: Every join-completion is the solution of a universal problem. J. Austral. Math. 17 (1974), 406–413.

    Article  MATH  Google Scholar 

  25. R. SIKORSKI: Boolean algebras. Springer-Verlag, Berlin-Göttingen-Heidelberg, 1960.

    MATH  Google Scholar 

  26. M. H. STONE: The theory of representations for Boolean algebras Trans. AMS 40 (1936), 37–111.

    Article  MATH  Google Scholar 

  27. P. V. VENKATANARASIMHAN: Pseudo-complements in posets. Proc. AMS 28 (1971), 9–15.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Mathematik, Universität Hannover, Welfengarten 1, 3000, Hannover 1, West Germany

    Marcel Erné & Gerhard Wilke

Authors
  1. Marcel Erné
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gerhard Wilke
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by Karl H. Hofmann

Rights and permissions

Reprints and permissions

About this article

Cite this article

Erné, M., Wilke, G. Standard completions for quasiordered sets. Semigroup Forum 27, 351–376 (1983). https://doi.org/10.1007/BF02572747

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02572747

Keywords

Search

Navigation