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How are called

monotone maps of partially ordered sets such that the image of a lower set is a lower set?

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 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.

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.

How are called

monotone maps of partially ordered sets such that the image of a lower set is a lower set?

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: The category of finite Esakia spaces is the category of partial orders with maps of this kind; the category of (all) Esakia spaces is dual to the category of Heyting algebras. I found this following Emil Jeřábek comment (thanks!). In modal logics these are known as p-morphisms.

Still, I would like to see terminology by poset people and a reference to classification.
https://en.wikipedia.org/wiki/Esakia_space

How are called

monotone maps of partially ordered sets such that the image of a lower set is a lower set?

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 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.

How are called

monotone maps of partially ordered sets such that the image of a lower set is a lower set?

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.

How are called

monotone maps of partially ordered sets such that the image of a lower set is a lower set?

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: The category of finite Esakia spaces is the category of partial orders with maps of this kind; the category of (all) Esakia spaces is dual to the category of Heyting algebras. I found this following Emil Jeřábek comment (thanks!). In modal logics these are known as p-morphisms.

Still, I would like to see terminology by poset people and a reference to classification.
https://en.wikipedia.org/wiki/Esakia_space