Double Affine Bruhat Order

We classify cocovers and covers of a given element of the double affine Weyl semigroup W with respect to the Bruhat order, specifically when W is associated to a finite root system that is irreducible and simply laced. We show two approaches: one extending the work of Lam and Shimozono, and its strengthening by Milicevic, where cocovers are characterized in the affine case using the quantum Bruhat graph of the finite Weyl group, and another, which takes a more geometrical approach by using the length difference set defined by Muthiah and Orr.