I am following a course on modal logic and I have issues with a specific area, namely filtrations that preserve frame properties. A definition for a filtration is given here: https://en.wikipedia.org/wiki/Kripke_semantics#Model_constructions. It gives a constraint for the relation R of the filtrated model but not the specific condition on when two equivalence classes in the filtrated model are related. This condition can be designed to make the filtrated model have certain properties like transitivity. For transitivity, I know which relation is used, namely:
$R^* XY \text{ iff } \mathcal{M}, s \vdash \square B \text{ implies } \mathcal{M}, t \vdash \square B \land B$ for all $s \in X$ and $t \in Y$. Where $R^*$ is the relation in the filtrated model. It turns out that this relation is transitive and reflexive when filtrating a transitive and reflexive model, preserving those conditions.
Now I need to make a filtration that preserves reflexivity, transitivity and also the condition: $\forall x \forall y \exists z (Rxz \land Ryz)$ when filtrating models that satisfy S4.2, being S4 + $\lozenge \square p \rightarrow \square \lozenge p$. I can't figure out how to define the relation $R^*$ in the filtration to preserve this property. My idea was to expand the transitive filtration but I can't figure it out.