Definition:Indexed Iterated Binary Operation

Definition
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:

Degenerate case
Let $\struct {G, *}$ be a unitary magma with identity $e$.

Let $a, b \in \Z$ be integers such that $a < b$.

Then:
 * $\ds \prod_{i \mathop = a}^b \map f i = e$

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

Also see

 * Definition:Indexed Summation


 * Definition:N-Ary Operation Induced by Binary Operation