For type SL2-hat, the quantum Bruhat graph formula for the Demazure product on the double affine Weyl semigroup is well-defined for positive levels and associative for levels greater than one.
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The Quantum Bruhat Graph for $\widehat{SL}_2$ and Double Affine Demazure Products
For type SL2-hat, the quantum Bruhat graph formula for the Demazure product on the double affine Weyl semigroup is well-defined for positive levels and associative for levels greater than one.