Definition:Iterated Binary Operation

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Definition

Indexed Iteration

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:

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


Iteration over Finite Set

Let $\struct {G, *}$ be a commutative semigroup.

Let $S$ be a finite non-empty set.

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

Let $n \in \N$ be the cardinality of $S$.

Let $g: \N_{<n} \to S$ be a bijection, where $\N_{<n}$ is an initial segment of the natural numbers.


The iteration of $*$ of $f$ over $S$, denoted $\displaystyle \prod_{s \mathop \in S} \map f s$, is the indexed iteration of $*$ of the composition $f \circ g$ over $\N_{<n}$:

$\displaystyle \prod_{s \mathop \in S} \map f s = \displaystyle \prod_{i \mathop = 0}^{n - 1} \map f {\map g i}$


Iteration over Set with Finite Support

Let $\struct {G, *}$ be a commutative monoid.

Let $S$ be a set.

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

Let the support $\operatorname {Supp} f$ be finite.


The iteration of $*$ of $f$ over $S$, denoted $\displaystyle \prod_{s \mathop \in S} \map f s$, is the iteration over the finite set $\operatorname{Supp} f$ of $f$:

$\displaystyle \prod_{s \mathop \in S} \map f s = \prod_{s \mathop \in \operatorname {Supp} f} \map f s$


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 $\closedint 1 n$ to $G$.

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


Also see