The orbit of closure-involution operations: the case of Boolean functions

For a set A of Boolean functions, a closure operator c and an involution i , let 𝒩_c,i(A) be the number of sets which can be obtained from A by repeated applications of c and i . The orbit 𝒪(c,i) is defined as the set of all these numbers. We determine the orbits 𝒪(S,i) where S is the closure defined by superposition and i is the complement or the duality. For the negation non , the orbit 𝒪(S, non ) is almost determined. Especially, we show that the orbit in all these cases contains at most seven numbers. Moreover, we present some closure operators where the orbit with respect to duality and negation is arbitrarily large.