Name of a group-like structure

This isn't an answer but it's too long for a comment. I suggest that $n-1$ is more important than $n$ in this context, for the following reason. Suppose $A$ is an $n$-group in a semigroup $S$. Then, for any positive integer $k$, we can define $A^k$ to be the set of all products of $k$ factors from $A$. The definition of $n$-group says $A^n\subseteq A=A^1$, and it follows that $A^k\subseteq A^r$ where $r$ is the remainder when $k$ is divided by $n-1$ (I take remainders to be in the range $1\leq r\leq n-1$ rather than the customary $0\leq r\leq n-2$ because there is no $A^0$).

The union $\bigcup_kA^k$ is a subsemigroup $S'$ of $S$. If the $A^k$'s for $1\leq k<n$ are pairwise disjoint, then we get a homomorphism from $S'$ to the additive group $\mathbb Z/(n-1)$ by sending all elements of $A^k$ to $k$. Conversely, any homomorphism $h$ from a subsemigroup of $S$ to $\mathbb Z/(n-1)$ gives an $n$-group, namely $h^{-1}\{1\}$. ("Conversely" may be an overstatement here, since the two processes are, in general, inverse to each other only on one side.) The situation where the $A^k$'s are not pairwise disjoint looks considerably more complicated, but maybe someone can provide some insight into it.