Iterated binary operation

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, an iterated binary operation is an extension of a <a href="/facts/Binary_operation/CYtow0bz">binary operation</a> on a <a href="/facts/Set_(mathematics)/BfucfHMq">set</a> S to a <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> on finite <a href="/facts/Sequence/gV2S4uqp">sequences</a> of elements of S through repeated application. Common examples include the extension of the <a href="/facts/Addition/apFWJBFB">addition</a> operation to the <a href="/facts/Summation/V2WUxoKn">summation</a> operation, and the extension of the <a href="/facts/Multiplication/VtcUjpNv">multiplication</a> operation to the <a href="/facts/Product_(mathematics)/aXY11nGS">product</a> operation. Other operations, e.g., the set-theoretic operations <a href="/facts/Union_(set_theory)/KVVXKfWl">union</a> and <a href="/facts/Intersection_(set_theory)/RPfUNJh9">intersection</a>, are also often <a href="/facts/Iterated/AMc1p7BL">iterated</a>, 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

∑
 ,
  
 ∏
 ,
  
 ⋃
 ,
 
 
 {\displaystyle \sum ,\ \prod ,\ \bigcup ,}
 
 and 
 
 
 
 ⋂
 
 
 {\displaystyle \bigcap }
 
, respectively.
More generally, iteration of a binary function is generally denoted by a slash: iteration of 
 
 
 
 f
 
 
 {\displaystyle f}
 
 over the sequence 
 
 
 
 (
 
 a
 
 1
 
 
 ,
 
 a
 
 2
 
 
 …
 ,
 
 a
 
 n
 
 
 )
 
 
 {\displaystyle (a_{1},a_{2}\ldots ,a_{n})}
 
 is denoted by 
 
 
 
 f
 
 /
 
 (
 
 a
 
 1
 
 
 ,
 
 a
 
 2
 
 
 …
 ,
 
 a
 
 n
 
 
 )
 
 
 {\displaystyle f/(a_{1},a_{2}\ldots ,a_{n})}
 
, following the notation for <a href="/facts/Fold_(higher-order_function)/gllDcCxw">reduce</a> in <a href="/facts/Bird%25E2%2580%2593Meertens_formalism/Us3fvK6D">Bird–Meertens formalism</a>.
In general, there is more than one way to extend a binary operation to operate on finite sequences, depending on whether the operator is <a href="/facts/Associative/EQX8lV6r">associative</a>, and whether the operator has <a href="/facts/Identity_element/9nss3xHY">identity elements</a>.

Iterated binary operation open-in-new

Iterated binary operation