Iterated binary operation

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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}).

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[edit]

Denote by aj,k, with j ≥ 0 and kj, the finite sequence of length kj of elements of S, with members (ai), for ji < k. Note that if k = j, the sequence is empty.

For f : S × S, define a new function Fl 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 has a unique left identity e, the definition of Fl can be modified to operate on empty sequences by defining the value of Fl on an empty sequence to be e (the previous base case on sequences of length 1 becomes redundant). Similarly, Fr can be modified to operate on empty sequences if f has a unique right identity.

If f is associative, then Fl equals Fr, 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 \prod_{i=0}^\infty a_i\, is defined, and equal to \lim\limits_{n\rightarrow\infty}\prod_{i=0}^na_i,  if and only if that limit exists.

Non-associative binary operation[edit]

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[edit]

External links[edit]