Let $A = P(\mathbb{N})$ be the powerset of the natural numbers. We can look at $A$ as the Boolean aglebra - having in mind the obvious operations on elements of $A$.
What I am interested in knowing is if perhaps the following holds:
If $f:A\mapsto A$ is an automorphism of $A$ then $f(\{x\}) = \{y\}$ where $x,y \in \mathbb{N}$
In other words, an automorphism of $A$ preserves singletons (see this for a definition of homomorphism ).
I was not able to find a proof of this fact but neither a counterexample.
Is anyone able to settle this question for me?