We consider nonmonotonic formula operators related to (logical) friendliness and investigate properties of some of them in comparison with classical consequence. We give an example of nonmonotonic noncompact operator, which is strictly less than classical consequence and strictly greater than friendliness operator. Also, we generalize the friendliness relation to that which is based on any Boolean algebra and show that each operator associated with such a relation is greater than or equal to that of friendliness. We show that in the last inequality relation the equality occurs, in particular, when a Boolean algebra is finite.