Dimensional operator


In mathematics, specifically set theory, a dimensional operator on a set E is a function from the subsets of E to the subsets of E.

Definition

If the power set of E is denoted P then a dimensional operator on E is a map
that satisfies the following properties for S,''T ∈ P'':
  1. S ⊆ d;
  2. d = d ;
  3. if S ⊆ T then d ⊆ d;
  4. if Ω is the set of finite subsets of S then d = ∪A∈Ωd;
  5. if x ∈ E and y ∈ d \ d, then x ∈ d.
The final property is known as the exchange axiom.

Examples

  1. For any set E the identity map on P is a dimensional operator.
  2. The map which takes any subset S of E to E itself is a dimensional operator on E.