I read that given a family of Boolean subalgebras, their union is not in general a Boolean algebra except if the family is directed. What is an example of a family of Boolean subalgebras whose union is not a Boolean algebra?

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    $\begingroup$ The union of the Boolean algebras $\{\varnothing,\{1\},\{2,3\},\{1,2,3\}\}$ and $\{\varnothing,\{2\},\{1,3\},\{1,2,3\}\}$ has six elements and is not a Boolean algebra. $\endgroup$
    – user14111
    Feb 26 at 1:34
  • $\begingroup$ That "except if the family is directed" is not quite right. If $B$ is a Boolean algebra with more than $4$ elements, then the family of all $4$-element subalgebras of $B$ is not directed but its union is a Boolean algebra, namely $B$. $\endgroup$
    – user14111
    Feb 26 at 1:43
  • $\begingroup$ @user14111 Next time please also pay attention to titles when making edits :) $\endgroup$ Feb 26 at 2:06


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