In Carter's book "Lie algebras of finite and affine type", he defines the $O$ category for Kac Moody algebras as follows:
I do not understand in what sense $\lambda<\lambda_i$ on the last part of the definition: I thought it just meant that $\lambda(h)<\lambda_i(h)$ for all $h\in H$, so that we were demanding the set of weights to have a supremum with respect to this relation $<$. But in that case one would always only need a single $\lambda_i$ (i.e. $s=1$), so I highly doubt that my interpretation is correct.
What am I getting wrong?