Twisted weak orders of Coxeter groups (Q2279683)

The author investigates the twisted weak order associated to a twisted Bruhat order for a Coxeter group. He explores the relationship between the lattice properties of such orders and the infinite reduced words. The main results that he obtains are the following theorems. 3.11. \((W,{\leq}'_B)\) and \((W,\leq_B)\) define the same order. 4.5. Take \(w \in W_l\) (the set of infinite reduced words). Then \((W, \leq_{\Phi_w})\) is a non-complete meet semilattice and \((W , \leq_{\Phi^+ \backslash \Phi_w} )\) is a non-complete join semilattice. 4.12. Let \(\widetilde{W}\) be an affine Weyl group. The following statements about a biclosed (resp. biconvex) set \(B\) in \(\widetilde{\Phi}^+\) are equivalent: (1) \(B\) is the inversion set of an infinite reduced word; (2) \((\widetilde{W} , \leq_B)\) is a non-complete meet semilattice.