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Are there any interesting applications of the structures $(A;B;f)$ consisting two posets $A$ and $B$ and a function $f:A\rightarrow B$? (consider also it special cases such as $A$ and/or $B$ being lattices, complete lattices; $f$ being monotone, preserving arbitrary meets and/or joins, etc.)

One interesting definition (to set an example) is for a given element $a\in B$ the set of elements over $a$ defined as $\{c\in A | f(c)\ge a \}$.

An example: Let $A$ is the poset (ordered by set theoretic inclusion) of subsets of some set $U$, let $B$ is the (ordered reverse to set theoretic inclusion) poset of filters on $U$ and $f$ sends a subset of $U$ into the corresponding principal filter. Then the elements over a filter would be just elements of this filter.

The special case when $f$ is an (implied) inclusion function of $A\subseteq B$ is considered in details in my draft article Filters on Posets and Generalizations.

An idea

I now have an idea (yet undeveloped, and I'm not sure it brings any interesting results).

Consider the following triples:

  • $(\mathsf{RLD}; \mathsf{FCD}; (\mathsf{FCD}))$;
  • $(\mathsf{FCD}; \mathsf{RLD}; (\mathsf{RLD})_{in})$;
  • $(\mathsf{FCD}; \mathsf{RLD}; (\mathsf{RLD})_{out})$

(see this my article for meaning of this notation).

There is the hope that these may prove infinite distributivity of these functions (over meets and/or joins of the relevant lattices) or something.

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Closure systems and convex structures come to mind.

If we let $\mathcal{P}(X)$ denote the collection of all subsets of the set $X$. Then a closure system for $X$ is a $\mathcal{C} \subseteq \mathcal{P}(X)$ satisfying, for all $\mathcal{A} \subseteq \mathcal{C}$ we have $\cap \mathcal{A} \in \mathcal{C}$. Define $$f\colon \mathcal{P}(X) \rightarrow \mathcal{C} \text{ as } f \colon A \mapsto \cap \{ C \in \mathcal{C} \colon A \subseteq C \} $$ where $A \subseteq X$. This is a closure system.

A convex structure has the additional property that for all $A \subseteq X$ we have $$f(A) = \cup \{ f(F) \colon F \text{ is a finite subset of }A \} .$$

A good reference for convex structures is van de Vel's book.

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