The canonical join complex for biclosed sets

Peter, Dillery,; Garver, Alexander

doi:10.48550/arxiv.1708.02580

Cited by 1 publication

(4 citation statements)

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“…Remark 5.3 A version of this poset of labels S has already been introduced in [7]. There the notion of segments plays the role of strings.…”

Section: Biclosed Sets and Biclosed Subcategoriesmentioning

confidence: 99%

“…There the notion of segments plays the role of strings. Many of the proofs [7] are applicable to the current work, and so we will frequently cite [7] in the sequel. We leave it to the reader to translate the relevant statements in terms of segments from [7] into statements in terms of strings in the current work.…”

Section: Biclosed Sets and Biclosed Subcategoriesmentioning

confidence: 99%

“…By definition, the set is closed so we show that J(w D ) is coclosed. The proof of [7,Lemma 3.6] implies that the set {w} D u∈D S(u) is coclosed. Thus, to complete the proof, we show that for any u ∈ J(w D )\ {w} D u∈D S(u) at least one element of each break of u belongs to J(w D ).…”

Section: Biclosed Sets and Biclosed Subcategoriesmentioning

confidence: 99%

“…From the second description, we obtain a large family of congruence-uniform lattices whose shard intersection orders are also lattices, which is not true for general congruence-uniform lattices; see [19] and [22,Problem 9.5]. The shard intersection orders we consider in this work include those discovered in [7] where the lattice property was also proved.…”

Section: Introductionmentioning

confidence: 99%

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A categorification of biclosed sets of strings

Garver

McConville

Mousavand³

2020

Journal of Algebra

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We consider the closure space on the set of strings of a gentle algebra of finite representation type. Palu, Pilaud, and Plamondon proved that the collection of all biclosed sets of strings forms a lattice, and moreover, that this lattice is congruence-uniform. Many interesting examples of finite congruence-uniform lattices may be represented as the lattice of torsion classes of an associative algebra. We introduce a generalization, the lattice of torsion shadows, and we prove that the lattice of biclosed sets of strings is isomorphic to a lattice of torsion shadows.Finite congruence-uniform lattices admit an alternate partial order known as the shard intersection order. In many cases, the shard intersection order of a congruence-uniform lattice is isomorphic to a lattice of wide subcategories of an associative algebra. Analogous to torsion shadows, we introduce wide shadows, and prove that the shard intersection order of the lattice of biclosed sets is isomorphic to a lattice of wide shadows.2010 Mathematics Subject Classification. 05E10, 16G20, 18B35. AG received support from the Canada Research Chairs Program and NSERC grant RGPIN/05999-2014. KM received Bourse de stage international du FRQNT for a 4-month research stay at MIT and was also partially supported by ISM Scholarship.

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“…Remark 5.3 A version of this poset of labels S has already been introduced in [7]. There the notion of segments plays the role of strings.…”

Section: Biclosed Sets and Biclosed Subcategoriesmentioning

confidence: 99%