\begin{definition}[Definition:Indexed Iterated Binary Operation]
Let $\struct {G, *}$ be a magma.
Let $a, b \in \Z$ be integers.
Let $\closedint a b$ be the integer interval between $a$ and $b$.
Let $f: \closedint a b \to G$ be a mapping.
The '''indexed iteration of $*$ of $f$ from $a$ to $b$''' is recursively defined and denoted:
:$\ds \prod_{k \mathop = a}^b \map f k = \begin {cases} \map f a & : b = a \\ \paren {\ds \prod_{k \mathop = a}^{b - 1} \map f k} * \map f b & : b > a \end {cases}$
For each ordered $n$-tuple $\tuple {a_1, a_2, \ldots, a_n} \in S^n$, the '''composite''' of $\tuple {a_1, a_2, \ldots, a_n}$ for $\oplus$ is the value at $\tuple {a_1, a_2, \ldots, a_n}$ of the $n$-ary operation defined by $\oplus$.
This '''composite''' is recursively defined and denoted:
{{begin-eqn}}
{{eqn | l = \bigoplus_{k \mathop = 1}^n a_k
      | r = \map {\oplus_n} {a_1, a_2, \ldots, a_n}
      | c = 
}}
{{eqn | r = \begin {cases} a & : n = 1 \\ \map {\oplus_m} {a_1, \ldots, a_m} \oplus a_{m + 1} & : n = m + 1 \end {cases}
      | c = 
}}
{{eqn | r = \paren {\paren {\cdots \paren {\paren {a_1 \oplus a_2} \oplus a_3} \oplus \cdots} \oplus a_{n - 1} } \oplus a_n
      | c = 
}}
{{end-eqn}}
\end{definition}