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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
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Okay as I was writing this I realized associativity is also required. All those four examples are associative.

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