Repeated application of an operation to a sequence
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
and , respectively.
More generally, iteration of a binary function is generally denoted by a slash: iteration of over the sequence is denoted by , 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.
^Saunders MacLane (1971). Categories for the Working Mathematician. New York: Springer-Verlag. p. 142. ISBN 0387900357.
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