Is there a name for 'groups' that only have a neutral element on the right and an inverse for each element on the right ?
If there is, does that name also hold for a neutral elt on the right and an inverse on the left ?
In any case, do those kind of 'groups' have special properties/are interesting to study ?
Yes, in the first case, they are called groups. See Right identity and Right inverse implies a group That is, the "only" is a priori, but it turns out in this case that the identity and inverses are $2$-sided.
(But for the case of having a right identity and left inverses, I do not know of a name or interesting properties. They need not be groups.)
