Is there an order-preserving surjective map $f: {\cal P}(\omega)/(fin) \to [0,1]$? Or from ${\cal P}(\omega)/(fin)$ onto $[0,1]\cap \mathbb{Q}$?



Order-preserving surjective map $f: {\cal P}(\omega)/(fin) \to [0,1]$
Replace $\omega$ with $X:=\mathbb Q\cap [0,1]$, and map each $A\subseteq X$ to $\limsup A\in [0,1]$. $\limsup$ is weakly increasing, and invariant under finite changes. This map is onto $[0,1]$.


If you want to map onto the dyadic rationals, partition $\omega$ into countably many infinite sets $\omega=A\_1\cup A\_2\cup A\_3\cup\cdots$, and map each set $B\subseteq \omega$ to 


* $\sum\_{n\in I\_B} 2^{-n}$, if $I\_B:=\{n: B \cap A\_n \mbox{ infinite}\}$ is finite
* $1$, if $I\_B$ is infinite.


If you want a map onto $\mathbb Q\cap [0,1]$, compose with some isomorphism.


