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(E) then a dimensional operator on E is a map
 * $$d:P(E)\rightarrow P(E)\,$$

that satisfies the following properties for S,T &isin; P(E):
 * 1) S &sube; d(S);
 * 2) d(S) = d(d(S)) (d is idempotent);
 * 3) if S &sube; T then d(S) &sube; d(T);
 * 4) if Ω is the set of finite subsets of S then d(S) = &cup;A&isin;Ωd(A);
 * 5) if x &isin; E and y &isin; d(S &cup; {x}) \ d(S), then x &isin; d(S &cup; {y}).

The final property is known as the exchange axiom.

Examples

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