Let $\kappa$ be an infinite cardinal. A family $\mathcal{A} \subseteq \mathcal{P}(\kappa)$ is independent if, for all $A_1,\ldots,A_n\in\mathcal{A}$ and $i_1,\ldots,i_n\in\{0,1\}$, we have
$$ \left|\bigcap_{j = 1}^n A_j^{i_j}\right| = \kappa $$
where $A^0 = A$ and $A^1 = \kappa\setminus A$.
Question: Is there an independent family $\mathcal{A}\subseteq\mathcal{P}(\kappa)$ such that the Boolean algebra generated by $\mathcal{A}$ and the subsets of $\kappa$ of size $< \kappa$ is all of $\mathcal{P}(\kappa)$?
I am particularly interested in the case $\kappa = \omega_1$, though an answer for any infinite $\kappa$ would be interesting.
This was answered on Math Overflow here. The gist is that an independent family of subsets of $\kappa$, along with the subsets of $\kappa$ of size $< \kappa$, can never generate all of $\mathcal{P}(\kappa)$ as a Boolean algebra.