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. More generally, iteration of a binary function is generally denoted by a slash: iteration of f over the sequence (a_{1}, a_{2} \ldots, a_{n}) is denoted by 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.

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. More generally, iteration of a binary function is generally denoted by a slash: iteration of f over the sequence (a_{1}, a_{2} \ldots, a_{n}) is denoted by 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. 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 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 a 0 , a 1 , a 2 , a 3 , … is an infinite sequence of real numbers, then the infinite product  \prod_{i=0}^\infty a_i is defined, and equal to \lim\limits_{n\to\infty}\prod_{i=0}^na_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. \mathop{\bigstar}_{k=1}^{n} a_k Meaning of symbols: Example: \mathop{\bigstar}_{k=1}^{4} a_k = \mathop{\bigstar}_{k=1}^{k=4} a_k = a_1 \star a_2 \star a_3 \star a_4 General form: \mathop{\bigstar}_{k \in K} a_k Restricted form: \mathop{\bigstar}_{1 \le k \le n \atop k \equiv 0 \pmod 2} a_k Infinite version: \mathop{\bigstar}_{k=1}^{\infty} a_k=\mathop{\bigstar}_{k=1}^{k\to\infty} a_k \mathop{\bigstar}_{k=n}^{n} a_k = a_n Expansion: \mathop{\bigstar}_{k=1}^{n} a_k = a_1 \star a_2 \star \dots \star a_n Recursion: \mathop{\bigstar}_{k=1}^{n} a_k = \left(\mathop{\bigstar}_{k=1}^{n-1} a_k\right)\star a_n Right recursion: \mathop{\bigstar}_{k=1}^{n} a_k = a_1 \star \left(\mathop{\bigstar}_{k=2}^{n} a_k\right) Splitting: \left(\mathop{\bigstar}_{k=1}^{m} a_k\right)\star\left(\mathop{\bigstar}_{k=m+1}^{n} a_k\right)=\mathop{\bigstar}_{k=1}^{n} a_k Permutation invariance: \mathop{\bigstar}_{k=1}^{n} a_k = \mathop{\bigstar}_{k=1}^{n} a_{\sigma(k)} Empty product (monoid): \mathop{\bigstar}_{k=1}^{0} a_k = e Idempotence: if a \star a = a , then \mathop{\bigstar}_{k=1}^{n} a = a Constant sequence: \mathop{\bigstar}_{k=1}^{n} a = a \star \cdots \star a Empty product = identity element Empty sum = 0 (in arithmetic monoids)

External links

Bulk action Parallel prefix operation Nuprl iterated binary operations

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