Bruhat Preclosure

Abstract: In 2011, Dyer published a series of conjectures on the weak order of Coxeter groups. One of these conjectures stated that the inversion set of the join of two elements in a Coxeter group is equal to some "closure" of the union of their inversion sets. In this paper we show that this "closure" is in fact a preclosure, which we call the Bruhat preclosure, but is a closure whenever our underlying set is an inversion set. By performing the Bruhat preclosure an infinite number of times we obtain a closure which we call the infinite Bruhat closure. We show in a uniform way that Dyer's conjecture is true when using the infinite Bruhat closure (instead of Bruhat preclosure) if the join exists between two elements. Finally, we end by showing in type A, the Bruhat preclosure is a closure thus giving a (second) proof that Dyer's conjecture is true in type A.