Definition:Iterated Binary Operation

Iteration over Set with Finite Support
Let $G$ be a commutative monoid.

Let $S$ be a set.

Let $f : S \to G$ be a mapping.

Note
Let $n\in\N$ be a natural number.

Note that an ordered $n$-tuple of elements of $G$ is by definition a mapping from the integer interval $\left[{1 \,.\,.\, n}\right]$ to $G$.

Thus the definition of indexed iterated binary operation includes the case of an ordered $n$-tuple.

Also known as
The indexed iterated binary operation of an ordered tuple is also known as their composite.

Also see

 * Definition:Summation
 * Definition:Product Notation (Algebra)