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 alsoisotone, 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?

In fact, 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). This may be relevant, considering that if $f$ is isotone, then $f^{-1}(\{y\})$ is a singleton for each $y \in T$.

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

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