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.