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| {{Unreferenced|date=April 2011}}
| | Hi there. Let me begin by introducing the author, her title is Sophia. To play lacross is some thing he would never give up. My wife and I reside in Mississippi but now I'm considering other choices. Invoicing is what I do.<br><br>My blog :: certified psychics; [http://203.250.78.160/zbxe/?document_srl=1792908 http://203.250.78.160/zbxe/?document_srl=1792908], |
| In [[mathematics]], an '''iterated binary operation''' is an extension of a [[binary operation]] on a set ''S'' to a [[function (mathematics)|function]] on finite [[sequence]]s 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 (mathematics)|product]] operation. Other operations, e.g., the set theoretic operations [[union (set theory)|union]] and [[intersection (set theory)|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
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| ::<math>\sum,\ \prod,\ \bigcup,</math> and <math>\bigcap</math>, respectively.
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| More generally, iteration of a binary function is generally denoted by a slash: iteration of <math>f</math> over the sequence <math>(a_{1}, a_{2} \ldots, a_{n})</math> is denoted by <math>f / (a_{1}, a_{2} \ldots, a_{n})</math>.
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| 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 element]]s.
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| == Definition ==
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| Denote by '''a'''<sub>''j'',''k''</sub>, with {{nowrap|''j'' ≥ 0}} and {{nowrap|''k'' ≥ ''j''}}, the finite sequence of length {{nowrap|''k'' − ''j''}} of elements of ''S'', with members (''a''<sub>i</sub>), for {{nowrap|''j'' ≤ ''i'' < ''k''}}. Note that if {{nowrap|''k'' {{=}} ''j''}}, the sequence is empty.
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| For {{nowrap|''f'' : ''S'' × ''S''}}, define a new function ''F''<sub>''l''</sub> on finite nonempty sequences of elements of ''S'', where
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| :<math>F_l(\mathbf{a}_{0,k})=
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| \begin{cases}
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| a_0, &k=1\\
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| f(F_l(\mathbf{a}_{0,k-1}), a_k), &k>1
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| \end{cases}.</math>
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| Similarly, define
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| :<math>F_r(\mathbf{a}_{0,k})=
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| \begin{cases}
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| a_0, &k=1\\
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| f(a_0, F_r(\mathbf{a}_{1,k})), &k>1
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| \end{cases}.</math>
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| If ''f'' has a unique left identity ''e'', the definition of ''F''<sub>''l''</sub> can be modified to operate on empty sequences by defining the value of ''F''<sub>''l''</sub> on an empty sequence to be ''e'' (the previous base case on sequences of length 1 becomes redundant). Similarly, ''F''<sub>''r''</sub> can be modified to operate on empty sequences if ''f'' has a unique right identity.
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| If ''f'' is associative, then ''F''<sub>''l''</sub> equals ''F''<sub>''r''</sub>, and we can simply write ''F''. Moreover, if an identity element ''e'' exists, then it is unique (see [[Monoid]]).
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| 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 set]]s.
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| If ''S'' also is equipped with a [[Metric (mathematics)|metric]] or more generally with [[topology]] that is [[Hausdorff space|Hausdorff]], so that the concept of a [[limit of a sequence]] is defined in ''S'', then an ''[[Infinity|infinite]] iteration'' on a countable sequence in ''S'' is defined exactly when the corresponding sequence of finite iterations converges. Thus, e.g., if ''a<sub>0</sub>'', ''a<sub>1</sub>'', ''a<sub>2</sub>'', ''a<sub>3</sub>'', ... is an infinite sequence of real numbers, then the [[infinite product]] <math>\prod_{i=0}^\infty a_i\,</math> is defined, and equal to <math>\lim\limits_{n\rightarrow\infty}\prod_{i=0}^na_i,</math> if and only if that limit exists.
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| ==Non-associative binary operation==
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| The general, non-associative binary operation is given by a [[magma (algebra)|magma]]. The act of iterating on a non-associative binary operation may be represented as a [[binary tree]].
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| ==See also==
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| *[[Fold (higher-order function)]]
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| *[[Infinite series]]
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| *[[Infinite product]]
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| *[[Continued fraction]]
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| ==External links==
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| * [http://web.archive.org/web/20071009181156/http://www.short-fuze.co.uk/~eddy/math/associate.html Bulk action] <!-- archived from the original -->
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| * [http://wotug.ukc.ac.uk/parallel/acronyms/hpccgloss/P.html#parallel%20prefix Parallel prefix operation]
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| * [http://www.cs.cornell.edu/Info/People/sfa/Nuprl/iterated_binops/Xiter_via_intseg_remark_INTRO.html Nuprl iterated binary operations]
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| [[Category:Binary operations]]
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Hi there. Let me begin by introducing the author, her title is Sophia. To play lacross is some thing he would never give up. My wife and I reside in Mississippi but now I'm considering other choices. Invoicing is what I do.
My blog :: certified psychics; http://203.250.78.160/zbxe/?document_srl=1792908,