Classification of Cocovers in the Double Affine Bruhat Order

We classify cocovers of a given element of the double affine Weyl semigroup $W_{\mathcal{T}}$ with respect to the Bruhat order, specifically when $W_{\mathcal{T}}$ is associated to a finite root system that is irreducible and simply laced. We do so by introducing a graphical representation of the length difference set defined by Muthiah and Orr and identifying the cocovering relations with the corners of those graphs. This new method allows us to prove that there are finitely many cocovers of each $x \in W_{\mathcal{T}}$. Further, we show that the Bruhat intervals in the double affine Bruhat order are finite.