The Dedekind-MacNeille completion as a reflector

Abstract

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.

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Communicated by K. Keimel

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Erné, M. The Dedekind-MacNeille completion as a reflector. Order 8, 159–173 (1991). https://doi.org/10.1007/BF00383401

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AMS subject classifications (1991)

  • 06A23
  • 18A40

Key words

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