Positive set-operators of low complexity

The powerset operator, P, is compared with other operators of similar type and logical complexity. Namely we examine positive operators whose defining formula has a canonical form containing at most a string of universal quantifiers. We call them ∀-operators. The question we address in this paper is: How is the class of ∀-operators generated? It is shown that every positive ∀-operator Γ such that Γ(∅) 6= ∅, is finitely generated from P, the identity operator Id, constant operators and certain trivial ones by composition, ∪ and ∩. This extends results of [3] concerning bounded positive operators. 2000 Mathematics Subject Classification. Primary 03E05, secondary 03E20.