How are called in combinatorics

monotone maps of partially ordered sets such that the image of a lower set is a lower set, i.e. closed (or open) maps of finite topologies? Is there a classification of such maps?

More precisely, I am interested in maps $f:X\rightarrow Y$ of preorders such that (i) $x\leq y$ implies $f(x)\leq f(y)$ (ii) $x\leq f(y)$ implies there is $x'\leq y$ such that $x=f(x')$.

Is there a classification of such maps between finite preorders? Is there a name for them?

The motivation for the question is that these are equivalent to closed maps of finite topological spaces.

Update: p-morphism is a name for these kind of maps in the theory of Kripke frames and modal logics (thanks to Emil Jeřábek). Esakia morphism and Esakia spaces is a name for something similar in the theory of Heyting algebras, but Esakia spaces are usually infinite.