We will show that if the cofinality of the ideal of Lebesgue measure zero sets is equal to \omega_1 then there exists a Boolean algebra B of cardinality \omega_1 which is not a union of strictly increasing \omega-sequence of its subalgebras. This generalizes a result of Just and Koszmider who showed that it is consistent with ZFC+\neg CH that such an algebra exists.

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**Last modified August 22, 2002.**