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Answer by David Gao for Whether an isotone bijection from a power set lattice...
Here is a more general fact:Proposition: Any monotone bijection $f: A \to B$ between two Boolean algebras is an isomorphism.Claim:$f(0) = 0$ and $f(1) = 1$.Proof of Claim: For all $a \in A$, $0 \leq a...
View ArticleAnswer by Emil Jeřábek for Whether an isotone bijection from a power set...
Yes, such a mapping necessarily sends singletons to singletons.Let $f\colon\mathcal P(S)\to\mathcal P(T)$ be a monotone bijection (or more generally, a surjective strictly monotone function). By...
View ArticleWhether an isotone bijection from a power set lattice to another sends...
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...
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