Given a binary operation $*:M\times M\rightarrow M$. I define the extension of $*$ to the subset of $M$ in the usual way (with $A,B \subseteq M$)
$a*A:=\{a\}*A$and $A*a:=A*\{a\}$,
$$A*B:=\bigcup_{a\in A,b\in B}\{a*b\}$$
$$ \forall \, c \in M, \; \forall k \in \mathbb{N} \; : \; M \underbrace{* (M * (M * \cdots *(M *}_{k \text{ } *\text{s}} \{c\}))) \subset M * \{c\} $$
Has this property got a name?
Are there known non-commutative (and/or non-associative) algebraic structures with this property?