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Encyclopedia > 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. Euclid, a famous Greek mathematician known as the father of geometry, is shown here in detail from The School of Athens by Raphael. ... In mathematics, a binary operation is a calculation involving two input quantities, in other words, an operation whose arity is two. ... Partial plot of a function f. ... In mathematics, a sequence is a list of objects (or events) arranged in a linear fashion, such that the order of the members is well defined and significant. ... 3 + 2 with apples, a popular choice in textbooks Addition is the basic operation of arithmetic. ... Sum redirects here. ... In mathematics, multiplication is an arithmetic operation which is the inverse of division, and in elementary arithmetic, can be interpreted as repeated addition. ...


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. In mathematics, associativity is a property that a binary operation can have. ... In mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a binary operation on that set. ...


For f : S × S, define a new function Fl on nonempty sequences of elements of S, where Fl((ai : 0 <= i < k)) = a0, if k = 1, or f(Fl((ai : 0 <= i < k-1)), ak), if k > 1. Similarly, define Fr((ai : 0 <= i < k)) = a0, if k = 1, or f(a0, Fr((ai : 1 <= i < k))), if k > 1.


If f is associative, Fl = Fr.


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.


See also

In mathematics, a binary operation is a calculation involving two input quantities, in other words, an operation whose arity is two. ... Sum redirects here. ...

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