×

Two consistency results concerning thin-tall Boolean algebras. (English) Zbl 0571.03022

A superatomic BA is one with all quotients atomic. Given such a BA A, let \(A=A_ 0,A_ 1,...,A_{\alpha}\), \(A_{\alpha}=\{0\}\), be the sequence obtained by successively dividing by the ideal of atoms, doing the natural thing at limit steps. Then \(\alpha +1\) is the height of A, and the width of A is the supremum of the cardinality of the atoms of \(A_{\beta}\), \(\beta\leq \alpha\). The result proved is: \(Con(ZFC+\neg CH\) \(+\) there is no superatomic BA of height \(\omega_ 2+1\) and width \(\omega\), and also no superatomic BA \(A\subseteq P_{\omega}\) such that height(A)\(\geq \omega_ 1+1\), \(A_{\beta}\) has \(\leq \omega_ 1\) atoms for \(\beta <\omega_ 1\), while \(A_{\omega_ 1}\) has \(\omega_ 2\) atoms). The forcing is Cohen forcing for adding \(\omega_ 2\) reals to a model of CH \(+\) ”There is a Kurepa tree”.
Reviewer: J.Monk

MSC:

03E35 Consistency and independence results
06E05 Structure theory of Boolean algebras
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] G. W. Day,Superatomic Boolean algebras, Pacific J. Math.23 (1967), 479-489. · Zbl 0161.01402
[2] I. Juh ász, K. Kunen andM. E. Rudin,Two more hereditarily separable non-Lindelöf spaces, Canad. J. Math.28 (1976), 998-1005. · Zbl 0336.54040 · doi:10.4153/CJM-1976-098-8
[3] I. Juhász andW. Weiss,On thin-tall scattered spaces, Coll. Math.40 (1978), 63-68. · Zbl 0416.54038
[4] K.Kunen,Set theory, North-Holland, 1980.
[5] M. Rajagopalan,A chain compact space which is not strongly scattered, Israel J. Math.23 (1976), 117-125. · Zbl 0331.54012 · doi:10.1007/BF02756790
[6] M.Weese,On the classification of compact scattered spaces, preprint. · Zbl 0464.54044
[7] M.Weese,On cardinal sequences of Boolean algebras, to appear in Algebra Universalis. · Zbl 0469.06005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.