Iterated binary operation

In

, 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

${\displaystyle \sum ,\ \prod ,\ \bigcup ,}$ and ${\displaystyle \bigcap }$, respectively.

More generally, iteration of a binary function is generally denoted by a slash: iteration of ${\displaystyle f}$ over the sequence ${\displaystyle (a_{1},a_{2}\ldots ,a_{n})}$ is denoted by ${\displaystyle f/(a_{1},a_{2}\ldots ,a_{n})}$, following the notation for reduce in Bird–Meertens formalism.

In general, there is more than one way to extend a binary operation to operate on finite sequences, depending on whether the operator is

.

Definition

Denote by 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 → S, define a new function F_l on finite nonempty sequences of elements of S, where $${\displaystyle F_{l}(\mathbf {a} _{0,k})={\begin{cases}a_{0},&k=1\\f(F_{l}(\mathbf {a} _{0,k-1}),a_{k-1}),&k>1.\end{cases}}}$$

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

If f 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 associative, then F_l equals F_r, and we can simply write F. Moreover, if an identity element e exists, then it is unique (see Monoid).

If f is

.

If S also is equipped with a

 ${\textstyle \prod _{i=0}^{\infty }a_{i}}$ is defined, and equal to ${\textstyle \lim \limits _{n\to \infty }\prod _{i=0}^{n}a_{i},}$ 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.

Iterated binary operations are used to represent an operation that will be repeated over a set subject to some constraints. Typically the lower bound of a restriction is written under the symbol, and the upper bound over the symbol, though they may also be written as superscripts and subscripts in compact notation. Interpolation is performed over positive integers from the lower to upper bound, to produce the set which will be substituted into the index (below denoted as i ) for the repeated operations.

Common notations include the big Sigma (repeated sum) and big Pi (repeated product) notations.

$${\displaystyle \sum _{i=0}^{n-1}i=0+1+2+\dots +(n-1)}$$ $${\displaystyle \prod _{i=0}^{n-1}i=0\times 1\times 2\times \dots \times (n-1)}$$

It is possible to specify set membership or other logical constraints in place of explicit indices, in order to implicitly specify which elements of a set shall be used:

$${\displaystyle \sum _{x\in S}x=x_{1}+x_{2}+x_{3}+\dots +x_{n}}$$

Multiple conditions may be written either joined with a

or separately:

$${\displaystyle \sum _{(i\in 2\mathbb {N} )\wedge (i\leq n)}i=\sum _{\stackrel {i\in 2\mathbb {N} }{i\leq n}}i=0+2+4+\dots +n}$$

Less commonly, any

(${\displaystyle \oplus }$) or

set union

(${\displaystyle \cup }$) may also be used.[2] For example, if S is a set of logical propositions:

$${\displaystyle \bigwedge _{p\in S}p=p_{1}\wedge p_{2}\wedge \dots \wedge p_{N}}$$

which is true

all of the elements of S are true.

• Unary operation
• Unary function
• Binary operation
• Binary function
• Ternary operation

External links

• Bulk action
• Parallel prefix operation Archived 2013-06-03 at the Wayback Machine
• Nuprl iterated binary operations