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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.

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  • $\begingroup$ Considered as maps of the upside-down Kripke frames $(X,\ge)\to(Y,\ge)$, such maps $f$ are exactly the p-morphisms (aka bounded morphisms). That’s likely not how order theorists would call them. $\endgroup$
    – Emil Jeřábek
    Commented Nov 2, 2016 at 20:42
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    $\begingroup$ This is a duplicate of my question mathoverflow.net/questions/137172/…; I'd still like an answer $\endgroup$
    – Benjamin Steinberg
    Commented Nov 2, 2016 at 21:58
  • $\begingroup$ ... from a poset person $\endgroup$
    – Benjamin Steinberg
    Commented Nov 2, 2016 at 22:12
  • $\begingroup$ @Emil: thanks! is there a classification or theory of p-morphisms between finite Kripke frames? $\endgroup$
    – user97621
    Commented Nov 2, 2016 at 22:45

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