While it is true that $\mathcal P(\kappa)$ is a complete Boolean algebra, it is not necessarily true that $\mathcal P(\kappa)/I$ is complete for an ideal $I$. In particular if we consider $I=J_{bd}$ the ideal of bounded subsets of $\kappa$.

For example, Hausdorff showed that there is an $(\omega_1,\omega_1)$ gap in $\mathcal P(\omega)/\textrm{Fin}$. That is, there is a sequence of order type $\omega_1+\omega_1^*$ which is strictly increasing, but there is no $A$ which realizes the cut between the $\omega_1$ and $\omega_1^*$ parts. Consider now the lower $\omega_1$ sets, if there was a supremum to this collection, it would have to be an element realizing the gap.

Assuming $V=L$. How complete is $\mathcal P(\kappa)/J_{bd}$? If the answer is strictly less than $\operatorname{cf}(\kappa)$, what if we limit ourselves to strictly increasing sequences? Can we get $\operatorname{cf}(\kappa)$-completeness?

That is, assuming that $\gamma<\kappa^+$ and $\langle A_\alpha\subseteq\kappa\mid\alpha<\gamma\rangle$ is strictly increasing modulo $J_{bd}$. Is there a least upper bound to the sequence?

I'd be interested to know (at least some partial) answer outside $V=L$, although that is less important.