Definition:Iterated Binary Operation

Magma
Let $\left({G, \oplus}\right)$ be a magma.

Let $a, b \in \Z$ be integers.

Let $\left[{a \,.\,.\, b}\right]$ be the integer interval between $a$ and $b$.

Let $f : \left[{a \,.\,.\, b}\right] \to G$ be a mapping.

For each ordered $n$-tuple $\left({a_1, a_2, \ldots, a_n}\right) \in S^n$, the composite of $\left({a_1, a_2, \ldots, a_n}\right)$ for $\oplus$ is the value at $\left({a_1, a_2, \ldots, a_n}\right)$ of the $n$-ary operation defined by $\oplus$.

This composite is recursively defined and denoted:

Unitary Magma
If the tuple is empty, then the composite is assigned the value of the identity of the operation (if this is a structure with an identity, that is):


 * $\oplus_0 \left({\varnothing}\right) = e_S$

Iteration over Finite Set
Let $G$ be a commutative semigroup.

Let $S$ be a finite nonempty set.

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

Commutative Monoid
Let $G$ be a commutative monoid.

Let $S$ be a nonempty set.

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

Iteration over Set with Finite Support
Let $G$ be a commutative monoid.

Let $S$ be a set.

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

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 see

 * Definition:Summation
 * Definition:Product Notation (Algebra)