Let be a Coxeter group and let be the positive roots. A subset of is called "biclosed" if, whenever we have roots , and with , if and then and, if and , then . The finite biclosed sets are the inversion sets of the elements of , and the containment between finite inversion sets is the weak order on . Dyer suggested studying the poset of all biclosed subsets of , ordered by containment, and conjectured that it is a complete lattice. As progress towards Dyer's conjecture, we classify all biclosed sets in the affine root systems. We provide both a type uniform description, and concrete models in the classical types , , , . We use our models to prove that biclosed sets form a complete lattice in types and , and we classify which biclosed sets are separable and which are weakly separable.
Mathematics Subject Classifications: 20F55, 17B22, 06B23
Keywords: Coxeter groups, root systems, affine Coxeter groups, lattice theory