Canonical extensions of posets

Morton, Wilmari

doi:10.1007/s00012-014-0292-1

Cited by 14 publications

(7 citation statements)

References 25 publications

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“…The differences between the lattice and poset cases arise from the fact that definitions for filters and ideals which are equivalent for lattices are not so for posets. This issue is discussed in detail in [24]. One way to address this systematically is to talk about the canonical extension of P with respect to F and I, where F and I are sets of 'filters' and 'ideals' of P respectively.…”

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%

“…This is a consequence of ambiguities surrounding the notions of 'filters' and 'ideals' in the more general setting. See [24] for a thorough investigation of this issue.…”

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confidence: 99%

Order polarities

Egrot

2019

Preprint

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“…Since taking MacNeille completions does distribute over products, as can be observed from the characterization of the MacNeille completion as the unique meet-and join-completion, if canonical amalgamations were to distribute over products then so would taking canonical extensions, and this is known not to be the case (see e.g. [14,Example 5.13]).…”

Section: Duals and Productsmentioning

confidence: 99%

“…As discussed in [3,Remark 2.3], the meanings of the terms 'filter' and 'ideal' are important here, as definitions that are equivalent for lattices diverge in the more general setting. The effect of varying these definitions on the canonical extension construction is investigated in [14].…”

Section: Introductionmentioning

confidence: 99%

Amalgamating poset extensions and generating free lattices

Egrot¹

2019

Preprint

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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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“…This is encoded by the (in)equations: The logic defined by L SL and these (in)equalities is known as separation logic or the logic of bunched implication or the distributive Lambek calculus depending on the context, and we shall denote it as SL. These (in)equations are canonical (residuated maps and their residuals are very well-behaved under canonical extension, even in posets see [Mor14]). As was shown in [DP15a], the adjoint transformationδ SL has right-inverses at every A in DL, and L SL -algebras therefore have Jónsson-Tarski extensions.…”

Section: Jónsson-tarski Vs Canonical Extensionsmentioning

confidence: 99%