On cardinal sequences of Boolean algebras
Let A be a Boolean algebra and let At(A) denote the set of atoms of A. Let \(J^{(0)}=\{0\}\); if \(\alpha =\beta +1\) then \(J^{(\alpha)}\) denotes the ideal generated by \(J^{(\beta)}\) and all \(a\in A\) such that \(a/J^{(\beta)}\in At(A/J^{(\beta)})\). If \(\alpha\) is a limit ordinal then \(J^{(\alpha)}=\cup \{J^{(\beta)}: \beta <\alpha \}\). The height of A, h(A), is the least \(\alpha\) with \(J^{(\alpha +1)}=J^{(\alpha)}\). The cardinal sequence of A is the sequence \((| At^{(\alpha)}|:\) \(\alpha <h(A)).\) In this paper it is shown Theorem: Let \(\alpha <\omega_ 2\), \(\tau =(c_ i:\) \(i<\alpha +1)\) a cardinal sequence with (i) \(c_ 0=\omega\); \(c_{\alpha}=1\); (ii) \(\omega \leq c_ i\leq 2^{\omega}\) for each \(i<\alpha\); (iii) if cf i\(=\omega_ 1\), then \(c_ i\leq \omega_ 1;\) then there exists a superatomic Boolean algebra with cardinal sequence \(\tau\).
scientific article; zbMATH DE number 3943881