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For any Kac-Moody root data $\mathcal D$, D. Muthiah and D. Orr have defineda partial order on the semi-direct product $W^+$ of the integral Tits cone withthe vectorial Weyl group of $\mathcal D$, and a compatible $\mathbb Z$-valuedlength function. We classify covers for this order and show that this lengthfunction defines a $\mathbb Z$-grading of $W^+$, generalizing the case ofaffine ADE root systems and giving a positive answer to a conjecture of Muthiahand Orr.

Grading of affine Weyl semi-groups of Kac-Moody type