Pointwise operation

From Citizendium
Revision as of 12:37, 8 March 2009 by imported>David E. Volk (subpages)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to navigation Jump to search
This article is a stub and thus not approved.
Main Article
Discussion
Related Articles  [?]
Bibliography  [?]
External Links  [?]
Citable Version  [?]
 
This editable Main Article is under development and subject to a disclaimer.

In abstract algebra, pointwise operation is a way of extending an operation defined on an algebraic struture to a set of functions taking values in that structure.

If O is an n-ary operator on a set S, written in functional notation, and F is a set of functions from A to S, then the pointwise extension of O to F is the operator, also written O, defined on n-tuples of functions in F with value a function from A to S, as follows

In the common case of a binary operation , written now in operator notation, we can write

For specific operations such as addition and multiplication the phrases "pointwise addition", "pointwise multiplication" are often used to denote their pointwise extension.