Definition:Iterated Binary Operation

Also denoted as
Depending on the notation that is used for the binary operation, alternative notations for the iterated operation are often seen:
 * $+$ becomes $\sum$
 * $\cap$ becomes $\bigcap$
 * $*$ and $\cdot$ may become $\prod$
 * $\oplus$ becomes $\bigoplus$

and so on.

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)