Iterated binary operation

In mathematics, an iterated binary operation is an extension of a binary operation on a set S to a function on finite sequences of elements of S through repeated application. Common examples include the extension of the addition operation to the summation operation, and the extension of the multiplication operation to the product operation. Other operations, e.g., the set theoretic operations union and intersection, are also often iterated, but the iterations are not given separate names. In print, summation and product are represented by special symbols; but other iterated operators often are denoted by larger variants of the symbol for the ordinary binary operator. Thus, the iterations of the four operations mentioned above are denoted

$\sum ,\ \prod ,\ \bigcup ,$ and $\bigcap$, respectively.

In general, there is more than one way to extend a binary operation to operate on finite sequences, depending on whether the operator is associative, and whether the operator has identity elements.

Definition

Denote by $\mathbf {a} _{j,k}$, with j >= 0 and k >= j, the finite sequence of length $(k-j)$ of elements of S, with members (a_i), for $j<=i<k$. Note that if k = j, the sequence is empty.

For f : S × S, define a new function $F_{l}$ on finite nonempty sequences of elements of S, where

$F_{l}(\mathbf {a} _{0,k})={\begin{cases}a_{0},&k=1\\f(F_{l}(\mathbf {a} _{0,k-1}),a_{k}),&k>1\end{cases}}.$

Similarly, define

$F_{r}(\mathbf {a} _{0,k})={\begin{cases}a_{0},&k=1\\f(a_{0},F_{r}(\mathbf {a} _{1,k})),&k>1\end{cases}}.$

If f is associative, F_l = F_r.

If f has a unique left identity e, the definition of F_l can be modified to operate on empty sequences by defining the value of F_l on an empty sequence to be e (the previous base case on sequences of length 1 becomes redundant). Similarly, F_r can be modified to operate on empty sequences if f has a unique right identity.

If f is commutative and associative, then F_l can operate on any non-empty finite multiset by applying it to an arbitrary enumeration of the multiset. If f moreover has an identity element e, then this is defined to be the same as the value of F_l on an empty multiset. If f is idempotent, then the above definitions can be extended to finite sets.

If S also is equipped with a metric or more generally with any topology, whence the concept of a limit of a sequence is defined in S, then an infinite iteration on a countable sequence in S is defined exactly when the corresponding sequence of finite iterations converges to a unique limit. Thus, e.g., if a_1, a_2, a_3,... is a countable sequence of real numbers, then the infinite product  $\prod _{i=1}^{\infty }a_{i}\,$  is defined and equal to  $\lim \limits _{n\rightarrow \infty }\prod _{i=1}^{n}a_{i},$  if and only if that limit exists.

Non-associative binary operation

The general, non-associative binary operation is given by a magma. The act of iterating on a non-associative binary operation may be represented as a binary tree.

See also

• Fold (higher-order function)
• Infinite series
• Infinite product
• Continued fraction

External links

• Bulk action
• Parallel prefix operation
• Nuprl iterated binary operations