Wikipedia / Iterated binary operation
Wikipedia / 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. 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
, respectively.
, 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.
Definition
Denote by aj,k, with and , the finite sequence of length of elements of S, with members (ai), for . Note that if , the sequence is empty.
For , define a new function Fl on finite nonempty sequences of elements of S
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 if and only if that limit exists.
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.
Basic iterated operations
Notation
The iterated binary operation is written as:
Meaning of symbols:Example:
General form:
Restricted form:
Infinite version:
Properties
:
Single element:
Expansion:
Recursion:
Right recursion:
Splitting:
Permutation invariance:
Empty product (monoid):
Idempotence:
Constant sequence:
Identity element and empty set
is a monoid, then:
Empty product = identity element
Empty sum = 0 (in arithmetic monoids)
Computer science
In functional programming, iterated binary operations correspond to higher-order functions such as fold or reduce.