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arXiv · 2609.27253

Bounded joins of biclosed sets

Yibo Gao·Yulin Peng·Hanlin Xu
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Abstract

Dyer conjectured that the join of two biclosed sets of positive roots can be described by increasing Bruhat paths whose reflection labels belong to their union. We give a type-uniform proof of this conjecture for all finite Coxeter groups. More generally, for a $2$-coclosed set $C$ of positive roots contained in an inversion set, we show that its $2$-closure is an inversion set $I(w)$ and that the elements reachable from the identity using reflections labeled by roots in $C$ form exactly the set $[e,w]_B w^{-1}$. The proof combines Dyer's closure criteria with a root-selection argument.

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Yibo Gao, Yulin Peng, Hanlin Xu. 2026-09-23. Bounded joins of biclosed sets. https://arxiv.org/abs/2609.27253

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