Distributive and completely distributive lattice extensions of ordered sets

Morton, Wilmari; Alten, C. J. Van

doi:10.1142/s0218196718500248

Cited by 6 publications

(5 citation statements)

References 27 publications

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“…By making the definitions of 'filter' and 'ideal' weak enough, this allows all notions of the canonical extension of a poset to be treated in a uniform fashion. This is the approach taken in [25], for example. Definition 2.5.…”

Section: Extensions and Completionsmentioning

confidence: 99%

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Order polarities

Egrot

2019

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We define an order polarity to be a polarity (X, Y, R) where X and Y are partially ordered, and we define an extension polarity to be a triple (e X , e Y , R) such that e X : P → X and e Y : P → Y are poset extensions and (X, Y, R) is an order polarity. We define a hierarchy of increasingly strong coherence conditions for extension polarities, each equivalent to the existence of a pre-order structure on X ∪ Y such that the natural embeddings, ι X and ι Y , of X and Y , respectively, into X ∪ Y preserve the order structures of X and Y in increasingly strict ways. We define a Galois polarity to be an extension polarity where e X and e Y are meet-and join-extensions respectively, and we show that for such polarities there is a unique pre-order on X ∪ Y such that ι X and ι Y satisfy particularly strong preservation properties. We define morphisms for polarities, providing the class of Galois polarities with the structure of a category, and we define an adjunction between this category and the category of ∆ 1 -completions and appropriate homomorphisms. We formalize the theory of extension polarities and prove a duality principle to the effect that if a statement is true for all extension polarities then so too must be its dual statement.

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Section: Extensions and Completionsmentioning

confidence: 99%

“…Polarities in the special case where X and Y are sets of subsets of some common set Z, where the relation R is that of non-empty intersection, and which also satisfy some additional conditions, have been referred to as polarizations in the literature [35,25]. Polarizations play an important role in the construction of canonical extensions.…”

Section: 3mentioning

confidence: 99%

Order polarities

Egrot

2019

Preprint

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“…As noted in [3,Remark 2.3], this choice of definition for ideal and filter is somewhat arbitrary, and there are others that also agree with the lattice version as used in [8]. For example, [15] defines filters to be upsets closed under existing finite meets, and defines ideals dually. This paper also generalizes the definition of canonical extension by defining it relative to a set F of filters and a set I of ideals, provided the pair (F , I) satisfies certain conditions.…”

Section: Canonical Extensionsmentioning

confidence: 99%

“…Going further, it is not necessary to restrict to the sets of all filters and ideals, however they are defined, or even to the relation of non-empty intersection. Going down this path leads [15] to define canonical extensions relative to a choice of a set of filters and a set of ideals. If we abandon explicit reference to filters, ideals and non-empty intersection altogether, but keep the essential ingredients of the Galois connection construction, we arrive at the generality of ∆ 1 -completions [9].…”

Section: Introductionmentioning

confidence: 99%

Amalgamating poset extensions and generating free lattices

Egrot¹

2019

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Given a poset P , a meet-extension e X : P → X and a joinextension e Y : P → Y , we define the canonical amalgamation of e X and e Y as an abstraction of a certain natural poset structure defined on X ∪ Y , and equipped with embeddings of X and Y corresponding to inclusions. This is essentially the extension of P obtained by 'freely' combining e X and e Y . By making particular choices for e X and e Y , the canonical amalgamation is the 'intermediate structure' that appears in the construction of canonical extensions. Our characterization is based on a kind of universal property. We find sufficient conditions for an order-preserving map f : P → Q to admit a lift to the canonical amalgamation that preserves some amount of the meet and join structure parametrized by a pair of cardinals. Using this, we show that the free lattice generated by P while preserving certain meets and joins can be obtained by constructing a chain of canonical amalgamations then taking the colimit. The objects in this chain can be thought of as approximations to the free lattice, and have their own universal properties.

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“…A recent paper of Morton and van Alten [7] shows, among other things, that the canonical extension D σ of a bounded distributive lattice D is the free completely distributive lattice generated by D. This means D σ is completely generated by D and for any completely distributive lattice C and any bounded lattice homomorphism h : D → C, there is a complete lattice homomorphism h * : D σ → C extending h. The purpose of this note is to provide a short alternate proof of this result, and to show its utility in easily establishing results about completely distributive lattices, canonical extensions, and complete retractions, both known and new. We assume familiarity with basics of canonical extensions as given in the first 8 pages of [4].…”

Section: Introductionmentioning

confidence: 99%