TITLE: How complete is $\mathcal P(\kappa)/J_{bd}$?
QUESTION [5 upvotes]: 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.

REPLY [6 votes]: 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 $\mathcal{P}(cf(\kappa))/J_{bd}$.