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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?

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Non-commutatuve groups have this property, and I think as long as it's associative, any other structure will as well. – Jacob Bond Aug 9 '14 at 16:17
Not every property in mathematics must carry a name, only the most interesting ones. This one doesn't seem particularly deep. – Alex M. Aug 9 '14 at 17:05

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