maximal antichain

Question

I don't understand the definition of Jech (set theory) for "maximal antichain". Let $B$ a boolean algebra and $A$ a subalgebra of $B$. $W\subseteq A^+$ is a maximal antichain if $\sum W=1$ and $W$ antichain. As $A^+\subseteq B^+$ and $1\in A^+\cap B^+$, $W$ is also a maximal antichain in $B$ (is there a mistake here ?). But it is saying that $W$ need not be maximal in $B$ .... why ? Thanks

Answer 1

It is easy to check that $W$ is an antichain in $B$ as well. So we will have to find a way to violate $\sum W = 1$ in $B$. The preservation of finite suprema means we will need to reach for an infinite $W$. An example is below.

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Consider $B = \mathcal P(\Bbb N)$, and let $A$ be the subalgebra of finite subsets of $\Bbb N$ not containing zero and their complements:

$$A = \{S \subseteq \Bbb N: \text{$S$ finite and $0 \notin S$, or $S$ infinite and $0 \in S$}\}$$

Then the set of singletons of positive integers $W = \{\{n\}: n > 0\}$ is clearly an antichain in $A$, and in $A$, $\sum W = \Bbb N$, while in $B$, $\sum W = \Bbb N_{>0}$.