completeness and saturation

Question

Let $B$ a complete Boolean algebra. Suppose, for $\kappa$ cardinal, that $B$ is not $\kappa$-saturated. Then there exists a partition $W$ of $B$. Because of completeness, we have $B=\sum W\in B$. So $B$ is in an element of an element of $B$ ?

Answer 1

Yes. A partition is not a collection of subsets of $B$ as we commonly think of a partition. It is an antichain, it's a subset of $B$ with the property: $$u,v\in W\rightarrow u=v\ \text{ or }\ u\cdot v=0_B.$$

Since the Boolean algebra is complete, $\sum W$ is some element in $B$. If $W$ is a maximal antichain then $\sum W=1_B$.

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The term partition corresponds to a partition of $A$, or a subset of $A$, when we think about $\mathcal P(A)$ as a Boolean algebra.