On the cofinalities of Boolean algebras and the ideal of null sets. (Q1771905)

For an infinite Boolean algebra \(\mathcal B\) its cofinality \(\text{cof}(\mathcal B)\) is the least infinite cardinal \(\kappa \) such that \(\mathcal B\) is the union of a strictly increasing sequence of type \(\kappa \) of subalgebras of \(\mathcal B\). Let \(\mathcal N\) denote the ideal of Lebesgue measure zero subsets of the real numbers and \(\text{cof}(\mathcal N)\) its cofinality. The authors prove that if \(\text{cof}(\mathcal N)=\omega _1\) then there exists a Boolean algebra \(\mathcal B\) of cardinality \(\omega _1\) such that \(\text{cof}(\mathcal B)=\omega _1\). This generalizes a result by \textit{W.\ Just} and \textit{P.\ Koszmider} [ibid. 28, 138--149 (1991; Zbl 0719.03023)] showing that the existence of such an algebra \(\mathcal B\) is consistent with \(\text{ZFC}+\neg \text{CH}\).