Say we have a set S and a binary operation $\star$ under which S is closed. Is this enough for us to derive an arbitrary (possibly infinitary) operation $\star$ in which the order of operations carried out does not affect the final result, and is defined as:
$$ \star(a_0, a_1, a_2, \ldots) = a_0 \star a_1 \star a_2 \star \ldots \iff a_i \in S \land i \leq n $$
If not, what other requirements must S and $\star$ satisfy for the above identity to hold?
The examples for such operations include:
| Operation | Set | Closed? | Commutative? |
|---|---|---|---|
| Addition | Real numbers | Yes | Yes |
| Multiplication | Real numbers | Yes | Yes |
| Multiplication | Matrices | Yes | No |
| Concatenation | Sequences | Yes | No |
Okay as I was writing this I realized associativity is also required. All those four examples are associative.