Functional equations in total negation

It is known that the only topological invariants P for which anti(P) = anti (2)(P), anti() denoting Bankston's total negation operator, are those which are determined purely by the cardinality of the underlying point-set. We e xamine equations of the form anti(n)(P) = anti(n)(not P), reaching similar conclusions for n less than or equal to 2 but weaker ones for n greater tha n or equal to 3. A corresponding investigation for total negation within a constraint is initiated. 2000 Mathematics subject classification: 54A25, 54 B05, 54D99.