In this paper, we initiate the study of the twisted weak order associated to a twisted Bruhat order for a Coxeter group and explore the relationship between the lattice property of such orders and the infinite reduced words. We show that for a 2 closure biclosed set B in Φ+, the B-twisted weak order is a non-complete meet semilattice if B is the inversion set of an infinite reduced word and that the converse also holds in the case of affine Weyl groups.

中文翻译：

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Coxeter 群的扭曲弱阶

在本文中，我们开始研究与 Coxeter 群的扭曲 Bruhat 阶相关的扭曲弱阶，并探索此类阶的格特性与无限约简词之间的关系。我们证明，对于 Φ+ 中的 2 闭包双闭集 B，如果 B 是无限约简词的逆集，则 B 扭曲弱阶是非完全满足半格，并且在仿射 Weyl 的情况下，逆序也成立组。