What does the set of dominant integral elements in a Cartan sub algebra look like?

Question

I'm reading about the theorem of the highest weight: Irreducible finite dimensional representations of a complex semisimple Lie algebra (with a fixed Cartan sub algebra, $\frak{h}$ and choice of positive roots in the root system $\Delta(\frak g, h)$) correspond bijectively to elements $\lambda\in \frak{h}^\ast$ which are dominant (meaning $\langle\lambda, \alpha\rangle\geq 0$ for every positive element $\alpha \in \Delta$) and algebraically integral (meaning $\frac{2\langle \lambda, \alpha \rangle}{|\alpha|^2} \in \mathbb{Z}$ for all $\alpha \in \Delta$). What does this subset of $\frak{h}^\ast$ look like? Evidently it is a discrete semigroup. Does it have a natural basis?