Given an arbitrary set of normal modal logics $\mathcal{L}$, one can define their sum $\bigoplus \mathcal{L}$ (or $\bigoplus_{L \in \mathcal{L}} L$ if you prefer) to be the least normal modal logic containing their union $\bigcup_{L \in \mathcal{L}} L$.
A logic $L$ is strongly Kripke complete if whenever a set of formulas $\Gamma$ (over the full unimodal language, i.e. with infinitely many atomic propositions) is $L$-consistent then $\Gamma$ is satisfiable in a single point in a model on a frame of $L$.
Now I was wondering whether it is known whether the set of strongly Kripke complete (normal modal) logics is closed under taking these $\bigoplus$ sums.
I was wondering since I'd like to know whether the strongly Kripke complete logics form a complete lattice. They are not closed under arbitrary intersections (but are under finite ones). If they would be closed under $\bigoplus$, like for example the canonical logics are, then they would still form a complete lattice.
