# Pointwise operation

Revision as of 12:37, 8 March 2009 by imported>David E. Volk (subpages)

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.