I have managed to free some time for my Set theory quest and have almost concluded chapter 7 (filters and Boolean algebras) of Jech. At this point I am left with only two exercises that I don't fully understand. This question is about exercise 7.33:
If $B$ is a $\kappa$-complete, $\kappa$-saturated Boolean algebra, then $B$ is complete. Hint: It suffices to show that $\sum X$ exists for every open $X \subseteq B$. If $X$ is open, show that $\sum X = \sum W$, where $W \subseteq X$ is a maximal antichain in X.
I have solved the question under the assumption that every antichain $W$ in $B$ must satisfy $|W| < \kappa$ (1). Under this assumption and following the hint, $\sum W$ exists because $B$ is $\kappa$-saturated; also, $\sum W$ is upper-bounded by any upper bound on $X$; finally, $x \leq \sum W\, \forall x \in X$, for if not, $W$ is not maximal.
My question is whether assumption (1) it true (and if so, how to prove it). It seems reasonable to conclude this from the fact that $B$ is $\kappa$-saturated, but I don't quite see how, because $\kappa$-saturation merely means that there is no partition of size $\kappa$. For example, what prohibits $B$ from containing an antichain $W$ satisfying $|W| = \kappa$ and $\sum W \notin B$ (i.e., an antichain of size $\kappa$ that doesn't imply a partition)?
Every antichain $W$ in $B$ can be extended to a maximal antichain $V$. But a maximal antichain $V$ is a partition (if $a<1$ is an upper bound for $V$, then $-a$ is disjoint from every element of $V$ and so $V\cup\{a\}$ is an antichain, contradicting maximality). So if there is an antichain of cardinality at least $\kappa$, it can be extended to a partition which has size greater than or equal to $\kappa$.
Note that there seems to be an error in Jech's definition of "$\kappa$-saturated": it should mean that there is no partition $W$ such that $|W|\geq\kappa$ (which by the previous paragraph is equivalent to the nonexistence of an antichain of size $\kappa$), rather than $|W|=\kappa$. These are equivalent for complete Boolean algebras but not in general. For instance, if $B$ is the algebra of countable or cocountable subsets of $\aleph_2$, then $B$ is $\aleph_1$-complete and $\aleph_1$-saturated by Jech's definition (there is no partition of size $\aleph_1$, though there is one of size $\aleph_2$!) but $B$ is not complete.