Iterated Binary Operation - Definition

Definition

Denote by 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

Similarly, define

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 a₀, a₁, a₂, a₃, ... is an infinite sequence of real numbers, then the infinite product is defined, and equal to if and only if that limit exists.