Extended weak order for the affine symmetric group

The extended weak order on a Coxeter group $W$ is the poset of biclosed sets in its root system. In (Barkley-Speyer 2024), it was shown that when $W=\widetilde{S}_n$ is the affine symmetric group, then the extended weak order is a quotient of the lattice $L_n$ of translation-invariant total orderings of the integers. In this article, we give a combinatorial introduction to $L_n$ and the extended weak order on $\widetilde{S}_n$. We show that $L_n$ is an algebraic completely semidistributive lattice. We describe its canonical join representations using a cyclic version of Reading's non-crossing arc diagrams. We also show analogous statements for the lattice of all total orders of the integers, which is the extended weak order on the symmetric group $S_\infty$. A key property of both of these lattices is that they are profinite; we also prove that a profinite lattice is join semidistributive if and only if its compact elements have canonical join representations. We conjecture that the extended weak order of any Coxeter group is a profinite semidistributive lattice.
Comment: v1: Preliminary version, comments welcome