By the work of Paul Cohen (on the continuum hypothesis), one can neither prove nor disprove from the axioms of ZFC that a bijection $f$ from the power set $\mathcal{P}(S)$ of a set $S$ to the power set $\mathcal{P}(T)$ of a set $T$ implies a bijection from $S$ to $T$.

Question. What if $f$ is also isotone, meaning that if $X \subseteq Y \subseteq S$ then $f(X) \subseteq f(Y)$? More precisely, does an isotone $f$ send singletons to singletons? Or is this also independent from ZFC?

If $f$ is isotone, then $f^{-1}(\{y\})$ is a singleton for each $y \in T$. However, I don't know of a single example of an isotone bijection from a power set lattice to another (or even into itself) that is not an order isomorphism (namely, whose functional inverse is not isotone too).

It is not difficult to show that an isotone bijection from one complete lattice to another need not be an order isomorphism. So, I'm really asking whether power set lattices are special in this regard.