Categorical characterization of the MacNeille completion

Banaschewski, Bernhard; Bruns, Günter

doi:10.1007/bf01898828

Cited by 147 publications

(86 citation statements)

References 6 publications

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“…For H = all order-embeddings (that is, strong monomorphisms) we have LInj(H) = complete join-semilattices and join-preserving maps. Indeed, Bernhard Banaschewski and Günter Bruns proved in [6] that every complete (semi)lattice X is left Kan-injective w.r.t. H since for every order-embedding h : A −→ A ′ and every monotone f : A −→ X we have f /h given by…”

Section: Linj(h)mentioning

confidence: 99%

Kan injectivity in order-enriched categories

Adámek¹,

Sousa²,

Velebil³

2014

Math. Struct. Comp. Sci.

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Abstract. Continuous lattices were characterised by Martín Escardó as precisely the objects that are Kan-injective w.r.t. a certain class of morphisms. We study Kan-injectivity in general categories enriched in posets. An example: ω-CPO's are precisely the posets that are Kan-injective w.r.t. the embeddings ω ֒→ ω + 1 and 0 ֒→ 1.For every class H of morphisms we study the subcategory of all objects Kan-injective w.r.t. H and all morphisms preserving Kan-extensions. For categories such as Top 0 and Pos we prove that whenever H is a set of morphisms, the above subcategory is monadic, and the monad it creates is a Kock-Zöberlein monad. However, this does not generalise to proper classes: we present a class of continuous mappings in Top 0 for which Kan-injectivity does not yield a monadic category.Dedicated to the memory of Daniel M. Kan (1927Kan ( -2013

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Section: Linj(h)mentioning

confidence: 99%

Kan injectivity in order-enriched categories

Adámek¹,

Sousa²,

Velebil³

2014

Math. Struct. Comp. Sci.

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“…It is well known that the MacNeille completion N (P) of a poset P is, up to isomorphism, the only completion of P that is doubly dense [3]. In this case the elements of the corresponding closure systems I N (P) and F N (P) are exactly what is usually known in the literature as the normal ideals and the normal filters of P.…”

Section: -Completionsmentioning

confidence: 99%

Δ1-completions of a Poset

Gehrke

Jansana

Palmigiano

2011

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A join-completion of a poset is a completion for which each element is obtainable as a supremum, or join, of elements from the original poset. It is well known that the join-completions of a poset are in one-to-one correspondence with the closure systems on the lattice of up-sets of the poset. A 1 -completion of a poset is a completion for which, simultaneously, each element is obtainable as a join of meets of elements of the original poset and as a meet of joins of elements from the original poset. We show that 1 -completions are in one-to-one correspondence with certain triples consisting of a closure system of down-sets of the poset, a closure system of up-sets of the poset, and a binary relation between these two systems. Certain 1 -completions, which we call compact, may be described just by a collection of filters and a collection of ideals, taken as parameters. The compact 1 -completions of a poset include its MacNeille completion and all its join-and all

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“…The standard construction of the completion can be found in Birkhoff [4]. Note that as we wish to construct the completion, we cannot use Proposition 4.2 of [3], which assumes its existence.…”

Section: K Admits Injective Hullsmentioning

confidence: 99%