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.

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

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.

[edit] 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.