set closed under an operation
A set is said to be closed under some map , if maps elements in to elements in , i.e., . More generally, suppose is the -fold Cartesian product . If is a map , then we also say that is closed under the map .
In the first example, the operation is of the type . In the latter, pointwise multiplication is a map .
The second example illustrated the somewhat odd definition of this term. When a function is defined, its domain and codomain are part of its definition, so it’s a little odd to talk about whether the function is closed or not: if you know what the function is, then you should know whether or not it is closed. So in practice, the way the term is used is this: We have a set and we have a function . We are given a subset , and asked whether is closed under . In other words, there is a natural way to make into a function , by restriction; the question is, does the result always lie in ? This would mean that yields a function , which is usually what we want.
Occasionally the word is used in a potentially confusing way. For example, left ideals in a ring are supposed to be closed under addition (which is an example of what we just discussed) and left multiplication by arbitrary ring elements. What this last condition means is that for every in the ring, the left ideal should be closed under the function .
|Title||set closed under an operation|
|Date of creation||2013-03-22 13:49:49|
|Last modified on||2013-03-22 13:49:49|
|Last modified by||archibal (4430)|