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:

$\displaystyle \prod_{k \mathop = a}^b \map f k = \begin {cases} \map f a & : b = a \\ \paren {\displaystyle \prod_{k \mathop = a}^{b - 1} \map f k} * \map f b & : b > a \end {cases}$

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:

 $\displaystyle \bigoplus_{k \mathop = 1}^n a_k$ $=$ $\displaystyle \map {\oplus_n} {a_1, a_2, \ldots, a_n}$ $\displaystyle$ $=$ $\displaystyle \begin {cases} a & : n = 1 \\ \map {\oplus_m} {a_1, \ldots, a_m} \oplus a_{m+1} & : n = m + 1 \end {cases}$ $\displaystyle$ $=$ $\displaystyle \paren {\paren {\cdots \paren {\paren {a_1 \oplus a_2} \oplus a_3} \oplus \cdots} \oplus a_{n - 1} } \oplus a_n$

Degenerate case

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

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

Then:

$\displaystyle \prod_{i \mathop = a}^b \map f i = e$

Also known as

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