The Dedekind-MacNeille completion as a reflector


We introduce a special type of order-preserving maps between quasiordered sets, the so-called cut-stable maps. These form the largest morphism class such that the corresponding category of quasiordered sets contains the category of complete lattices and complete homomorphisms as a full reflective subcategory, the reflector being given by the Dedekind-MacNeille completion (alias normal completion or completion by cuts). Suitable restriction of the object class leads to the category of separated quasiordered sets and its full reflective subcategory of completely distributive lattices. Similar reflections are obtained for continuous lattices, algebraic lattices, etc.

This is a preview of subscription content, log in to check access.


  1. 1.

    A.Abian (1968) On definitions of cuts and completion of partially ordered sets, Z. Math. Logik Grundl. der Math. 14, 299–309.

    Google Scholar 

  2. 2.

    B.Banaschewski (1956) Hüllensysteme und Erweiterung von Quasi-Ordnungen, Z. Math. Logik Grundl. der Math. 2, 117–130.

    Google Scholar 

  3. 3.

    B.Banaschewski and G.Bruns (1967) Categorical characterization of the MacNeille completion, Arch. Math. (Basel) 43, 369–377.

    Google Scholar 

  4. 4.

    G. Birkhoff (1973) Lattice Theory, Amer. Math. Soc. Coll. Publ. 25, 3rd ed., Providence, R.I.

  5. 5.

    A.Bishop (1978) A universal mapping characterization of the completion by cuts. Algebra Universalis 8, 349–353.

    Google Scholar 

  6. 6.

    T. S.Blyth and M. S.Janowitz (1972) Residuation Theory, Pergamon Press, Oxford.

    Google Scholar 

  7. 7.

    G.Bruns (1962) Darstellungen und Erweiterungen geordneter Mengen I, II, J. Reine Angew. Math. 209, 167–200, and 210, 1–23.

    Google Scholar 

  8. 8.

    P.Crawley and R. P.Dilworth (1973) Algebraic Theory of Lattices, Prentice-Hall, Inc., Englewood Cliffs, N.J.

    Google Scholar 

  9. 9.

    R.Dedekind (1872) deStetigkeit und irrationale Zahlen, 7th ed. (1969), Vieweg, Braunschweig.

    Google Scholar 

  10. 10.

    M. Erné (1980) Verallgemeinerungen der Verbandstheorie, I, II, Preprint No. 109 and Habilitations-schrift, Institut für Mathematik, Universität Hannover.

  11. 11.

    M.Erné (1980) Separation axioms for interval topologies, Proc. Amer. Math. Soc. 79, 185–190.

    Google Scholar 

  12. 12.

    M.Erné (1981) A completion-invariant extension of the concept of continuous lattices. In: B.Banaschewski and R.-E.Hoffmann (eds.), Continuous Lattices, Proc. Bremen 1979, Lecture Notes in Math. 871, Springer-Verlag, Berlin-Heidelberg-New York, 43–60.

    Google Scholar 

  13. 13.

    M. Erné (1981) Scott convergence and Scott topology in partially ordered sets, II. In: Continuous Lattices, Proc. Bremen 1979 (see [12]), 61–96.

  14. 14.

    M.Erné (1982) Distributivgesetze und Dedekindsche Schnitte, Abh. Braunschweig. Wiss. Ges. 33, 117–145.

    Google Scholar 

  15. 15.

    M.Erné (1982) deEinführung in die Ordnungstheorie, Bibl. Inst. Wissenschaftsverlag, Mannheim.

    Google Scholar 

  16. 16.

    M.Erné (1983) Adjunctions and standard constructions for partially ordered sets. In: G.Eigenthaler et al. (eds.), Contributions to General Algebra 2, Proc. Klagenfurt Conf. 1982. Hölder-Pichler-Tempski, Wien, 77–106.

    Google Scholar 

  17. 17.

    M.Erné (1986) Order extensions as adjoint functors, Quaestiones Math. 9, 149–206.

    Google Scholar 

  18. 18.

    M.Erné (1987) Compact generation in partially ordered sets, J. Austral. Math. Soc. 42, 69–83.

    Google Scholar 

  19. 19.

    M. Erné (1988) The Dedekind-MacNeille completion as a reflector. Preprint No. 1183, Techn. Hochschule Darmstadt.

  20. 20.

    M. Erné (1988) Bigeneration and principal separation in partially ordered sets, Preprint No. 1185, Techn. Hochschule Darmstadt; (1991) Order 8, 197–221.

  21. 21.

    M. Erné (1989) Distributive laws for concept lattices, Preprint No. 1230, Techn. Hochschule Darmstadt.

  22. 22.

    O.Frink (1954) Ideals in partially ordered sets, Amer. Math. Monthly 61, 223–234.

    Google Scholar 

  23. 23.

    G.Gierz, K. H.Hofmann, K.Keimel, J. D.Lawson, M.Mislove, and D. S.Scott (1980) A Compendium of Continuous Lattices, Springer-Verlag, Berlin-Heidelberg-New York.

    Google Scholar 

  24. 24.

    M.Kolibiar (1962) Bemerkungen über Intervalltopologie in halbgeordneten Mengen. In: General Topology and Its Relations to Modern Analysis and Algebra, Proc. Sympos. Prague 1961, Academic Press, New York, 252–253.

    Google Scholar 

  25. 25.

    H. M.MacNeille (1937) Partially ordered sets, Trans. Amer. Math. Soc. 42, 416–460.

    Google Scholar 

  26. 26.

    G. N.Raney (1953) A subdirect union representation for completely distributive complete lattices, Proc. Amer. Math. Soc. 4, 518–522.

    Google Scholar 

  27. 27.

    J.Schmidt (1956) Zur Kennzeichnung der Dedekind-MacNeilleschen Hülle einer geordneten Menge, Arch. Math. (Basel) 7, 241–249.

    Google Scholar 

  28. 28.

    D. P.Strauss (1968) Topological lattices, Proc. London Math. Soc. 18, 217–230.

    Google Scholar 

  29. 29.

    R.Wille (1982) Restructuring lattice theory: an approach based on hierarchies of concepts. In: I.Rival (ed.), Ordered Sets, Reidel, Dordrecht-Boston, 445–470.

    Google Scholar 

Download references

Author information



Additional information

Communicated by K. Keimel

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Erné, M. The Dedekind-MacNeille completion as a reflector. Order 8, 159–173 (1991).

Download citation

AMS subject classifications (1991)

  • 06A23
  • 18A40

Key words

  • (Quasi-)ordered set
  • complete lattice
  • completion
  • cut
  • reflector