On interrelations between strongly, weakly and chord separated set-systems (a geometric approach)

Данилов, В. И.; Karzanov, Alexander V.; Koshevoy, Gleb A.

doi:10.1007/s10801-021-01047-5

Cited by 1 publication

(3 citation statements)

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“…We will also need the following concepts from [19]. Given a cubillage Q of Z(n, δ + 1) and a subcomplex M of Q, we say that M is a membrane in Q if M is a cubillage of Z(n, δ).…”

Section: Separated Collectionsmentioning

confidence: 99%

“…This means that the analogues of the map g for other crosssections of cubillages are not generally surjective. In [19,20,21], rather than studying the intersection of a cubillage with these hyperplanes, the fragmentation of a cubillage into different pieces cut by these hyperplanes is studied.…”

Section: Cubillagesmentioning

confidence: 99%

“…Our approach is to use the description of the higher Bruhat orders in terms of cubillages of cyclic zonotopes and the description of these objects in terms of separated collections established in [32] and studied extensively in [18,19,16,20,21]. These tools allow us to construct cubillages which are pre-images under the map g, which is instrumental in the proof of Theorem 1.…”

Section: Introductionmentioning

confidence: 99%

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The First Higher Stasheff-Tamari Orders are Quotients of the Higher Bruhat Orders

Williams¹

2023

Electron. J. Combin.

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We prove the conjecture that the higher Tamari orders of Dimakis and Müller-Hoissen coincide with the first higher Stasheff-Tamari orders. To this end, we show that the higher Tamari orders may be conceived as the image of an order-preserving map from the higher Bruhat orders to the first higher Stasheff-Tamari orders. This map is defined by taking the first cross-section of a cubillage of a cyclic zonotope. We provide a new proof that this map is surjective and show further that the map is full, which entails the aforementioned conjecture. We explain how order-preserving maps which are surjective and full correspond to quotients of posets. Our results connect the first higher Stasheff-Tamari orders with the literature on the role of the higher Tamari orders in integrable systems.

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“…We will also need the following concepts from [19]. Given a cubillage Q of Z(n, δ + 1) and a subcomplex M of Q, we say that M is a membrane in Q if M is a cubillage of Z(n, δ).…”

Section: Separated Collectionsmentioning

confidence: 99%

Section: Cubillagesmentioning

confidence: 99%