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. ^[1] 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

${\displaystyle \sum ,\ \prod ,\ \bigcup ,}$ 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 associative, and whether the operator has identity elements.

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, 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 commutative and associative, then F 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 value of F 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 topology that is Hausdorff, so that 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. Thus, e.g., if a₀, a₁, a₂, a₃, … is an infinite sequence of real numbers, then the infinite product  ${\textstyle \prod _{i=0}^{\infty }a_{i}}$ 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.

Notation

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. 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.

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

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

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

Though binary operators including but not limited to exclusive or and set union may be used. ^[2]

Let S be a set of sets

$${\displaystyle \bigcup _{s\in S}s_{i}=s_{1}\cup s_{2}\cup \dots \cup s_{N}.}$$

Let S be a set of logical propositions ^{[ clarification needed ]}

$${\displaystyle \bigoplus _{s\in S}s_{i}=s_{1}\oplus s_{2}\oplus \dots \oplus s_{N}.}$$

Let S be a set of multivectors in a Clifford algebra/geometric algebra

$${\displaystyle \bigwedge _{s\in S}s_{i}=s_{1}\wedge s_{2}\wedge \dots \wedge s_{N},}$$

$${\displaystyle \prod _{s\in S}s_{i}=s_{1}s_{2}\dots s_{N}.}$$

Note how in the above, no upper bound is used, because it suffices to express that the elements ${\displaystyle s_{i}}$ are elements of the set S.

It is also to produce a repeated operation given a number of constraint joined by a conjunction (and), for example:

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

which may also be denoted

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

See also

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

References

1. Saunders MacLane (1971). Categories for the Working Mathematician. New York: Springer-Verlag. p. 142. ISBN   0387900357.
2. Weisstein, Eric W. "Union". mathworld.wolfram.com. Wolfram Mathworld. Retrieved 30 January 2018.

External links

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