I have recently been thinking about the following mathematical structure. I am wondering if this structure has been studied before or if it is new (I doubt it as it seems a fairly obvious structure to consider)
Let $A$ be a set. Let $\mathscr{A}$, be the set consisting of elements of the form $(a_1,a_2,...,a_n)$, where $n$ is a natural number or infinity, where $a_i \in A$. (Sorry this is worded badly), I means that the elements of $\mathscr{A}$ are finiite or infinite sequences of elements in $A$.
We define an interaction, $\times$, to be a function $C \rightarrow \mathscr{A}$ where $C$ is a subset of $\mathscr{A}$
-
Is there a name for this structure? The reason I think this structure is worth studying is because other algebraic structures, such as groups, rings and modules over a ring can be considered to be a particular type of the above structure.