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 $B$ ?
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I'd say it's $\top = \sum W$, where $\top$ is the unit of $B$. You may have mixed up an abstract Boolean algebra and one of subsets of a set (if $B$ is a BA of subsets of a set $X$, then $X \ne B$). – Lord_Farin May 24 '13 at 08:15
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@Lord_Farin: That's true if $W$ is a maximal partition. – Asaf Karagila May 24 '13 at 08:57
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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 $\sum W=1_B$.
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.
Asaf Karagila
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1Marc, it's a subset of $B$ which has the properties I mentioned. Partition and antichain are synonymous in this context. – Asaf Karagila May 24 '13 at 09:11
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Marc, we don't. That's just the "basic example" and it gives us motivation for the definition. – Asaf Karagila May 24 '13 at 14:08