Is there a standard name for order-preserving maps $f\colon P\to Q$ of posets with the property that the image of a lower set is a lower set, or equivalently if $q\leq f(p)$ then there exists $p'\leq p$ with $f(p')=q$?


If you view the poset as a category, then this condition says that the functor associated to $P$ is surjective on in-stars.


In my research I need to consider the category of posets with these kinds of morphisms and I would like to know their name. For example, if $P$ and $Q$ are face posets of regular CW complexes, then this property says that the image of each closed cell is a closed cell.



Terminology question for poset maps
I think one name for this is a [simulation](http://ncatlab.org/nlab/show/simulation).


