Does a bijection between the atoms of two atomistic posets induce a bijection between the corresponding ground sets?

Question

Recall that a poset $(X,\le)$ is said atomistic if every element may be expressed as (arbitrary) join of atoms. The most important case is the powerset lattice, where atoms are singletons. In such a case, notice that if there exists an order-preserving bijection between atoms, we can construct a bijection (an isomorphism) between the corresponding ground sets (in other terms, given an order-preseving map $F: \mathcal{P}(X) \to \mathcal{P}(Y)$ which maps bijectively singletons to singletons, we can construct a bijection from $X$ to $Y$ using $F$). Does a similar property hold for any atomistic poset?