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Monroe Eskew
Monroe Eskew
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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 essentially 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 $\mathcal{P}(cf(\kappa))/J_{bd}$.

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 essentially 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 $\mathcal{P}(cf(\kappa))/J_{bd}$.

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 $\mathcal{P}(cf(\kappa))/J_{bd}$.

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Monroe Eskew
Monroe Eskew
  • 18.6k
  • 5
  • 53
  • 114

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 essentially 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 $\mathcal{P}(cf(\kappa))/J_{bd}$.