Let $X$ be a set, and let $\text{Part}(X)$ denote the collection of all partitions of $X$. For $A, B\in \text{Part}(X)$ we set $A\leq B$ if $A$ refines $B$, that is for all $a\in A$ there is $b\in B$ such that $a\subseteq b$. This relation defines a partial order on $\text{Part}(X)$.


If $X$ is an infinite set, is there a surjective order-preserving map $f:{\cal P}(X)\to \text{Part}(X)$ (where ${\cal P}(X)$ denotes the power-set of $X$, ordered by $\subseteq$)?



Surjective order-preserving map $f:{\cal P}(X)\to \text{Part}(X)$
It may be clarifying to work with equivalence relations $E$ on $X$ rather than partitions on $X$. The two are in natural bijection, with $E$ inducing a partitioning quotient map $q: X \to X/E$, and $X/E$ refines $X/E'$ iff $E \subseteq E'$ as subsets of $X \times X$. 


Next, there is a surjective order-preserving map $P(X \times X) \to \text{Equiv}(X)$ where a general relation $R \in P(X \times X)$ is mapped to the equivalence relation $E\_R$ that it generates. This is clearly surjective since an equivalence relation $E$ is mapped to itself. 


Finally, if $X$ is infinite, there is a bijection $X \cong X \times X$, which induces an isomorphism of orders $P(X) \to P(X \times X)$. The composite 


$$P(X) \to P(X \times X) \to \text{Equiv}(X)$$ 


provides what you want. 


