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.



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> 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?
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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.



How complete is $\mathcal P(\kappa)/J_{bd}$?
If $\kappa$ is regular, then $\mathcal{P}(\kappa)/J\_{bd}$ is a $\kappa$-complete boolean algebra. If $\langle A\_\alpha : \alpha < \delta < \kappa \rangle$ is a sequence of subsets of $\kappa$, then the union of these is a least upper bound. This uses the $\kappa$-completeness of $J\_{bd}$. It is not $\kappa^+$-complete for the same reason Paul mentioned for $\omega$. Take any partition of $\kappa$ into $\kappa$ many unbounded sets, $\langle A\_\alpha : \alpha < \kappa \rangle$, let $B$ be an upper bound, and let $C \subseteq B$ have one point from each $A\_\alpha \cap B$ removed. So $C$ is an upper bound and strictly below $B$. For singular $\kappa$, the algebra is $cf(\kappa)$-complete but not $cf(\kappa)^+$-complete: Take any unbounded $A$ set of size $cf(\kappa)$ and consider the algebra below $A$. It is isomorphic to $\mathcal{P}(cf(\kappa))/J\_{bd}$.


