It is well-known that $(\text{Ord},+,0)$ and $(\text{Ord},\cdot,1)$ are monoids, but I haven't found references on these structures or other simpler ones (like $(\omega_1,+,0)$). For example, it would be useful for me to have references on submonoids, generators and similar things in these monoids.
Does anyone know of references that address this kind of structures? If there aren't too many references for this, is there a reason why this is happening? Maybe most of the interesting results about other monoids can be applied to ordinals or it is just that they are not interesting structures.