Definition:Iterated Binary Operation

Definition
Let $\left({S, \oplus}\right)$ be an algebraic structure.

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 can be defined and denoted:

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):


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

Also see

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