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Erné, Marcel

The Dedekind-MacNeille completion as a reflector. (English) Zbl 0738.06004

Order 8, No. 2, 159-173 (1991).
Summary: 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.

Cited in 1 Review
Cited in 38 Documents

MSC:

06B23 Complete lattices, completions
18A40 Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.)
06A06 Partial orders, general

Keywords:

order-preserving maps; cut-stable maps; category of quasiordered sets; category of complete lattices and complete homomorphisms; full reflective subcategory; reflector; Dedekind-MacNeille completion; completion by cuts; completely distributive lattices; continuous lattices; algebraic lattices
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\textit{M. Erné}, Order 8, No. 2, 159--173 (1991; Zbl 0738.06004)
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References:

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