PreprintCC0arXiv
Bounded joins of biclosed sets
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.
Related
- On inversion sets of joins in weak Bruhat order
- BiLipschitz and bounded displacement equivalence of Delone sets
- Quasi-overlap and quasi-grouping functions on bounded pseudo-ordered sets
- Improved Algorithms for Beck--Fiala with Bounded Sets
- Surjective Hausdorff Isometries of Hyperspaces of Bounded Closed Convex Sets
BibTeX
@misc{gao2026bounded,
title = {{Bounded joins of biclosed sets}},
author = {Yibo Gao and Yulin Peng and Hanlin Xu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.27253},
eprint = {2609.27253},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27253},
note = {CC0},
}
RIS
TY - UNPB
TI - Bounded joins of biclosed sets
AU - Yibo Gao
AU - Yulin Peng
AU - Hanlin Xu
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.27253
UR - https://arxiv.org/abs/2609.27253
AB - 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.
ER -