Is it possible for a social choice rule to simultaneously violate the condition of non dictatorship and weak pareto property? Non dictatorship is when there is no individual such that if he prefers x over y, then so does the society irrespective of what the society feels. And Pareto property is that if all individuals prefer x over y, then so should the society. It is about how society preserves total agreement.
I don't think it's possible because of the following reason. Now, if we have dictatorship and rest all also agree with him, then society is essentially preserving that opinion (so Pareto property satisfied). On the other hand, if Pareto property is not satisfied, ie total agreement of x preferred over y by each individual doesn't not lead to society choosing x over y, this means there is no individual such that if he prefers x over y, then society prefers x over y. So non dictatorship is satisfied. Please see if this line of thought is correct.