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Cardinal functions on Boolean algebras. (English) Zbl 0706.06009

Lectures in Mathematics, ETH Zürich. Basel etc.: Birkhäuser Verlag. 152 p. sFr 28.00; DM 32.00; $ 18.50 (1990).
This fine monograph will be the standard reference on cardinal invariants in Boolean algebras for years to come. Each chapter focuses on one invariant, along with variations of it. Notation has been cleaned up so that parallels between Boolean algebraic and topological invariants are obvious. Summarizing charts are included, as well as a long list of open problems. While most of the results use no extra axioms, quite a bit of set theoretical sophistication is needed to follow the arguments, which have a strongly combinatorial flavor. CH and \(\diamondsuit\) are used unashamedly, and forcing results are only referred to, with no proofs given. There is also a good deal of reference to topology. It is assumed that the reader has read Volume I of the Handbook of Boolean Algebra (1989; Zbl 0671.06001) which is S. Koppelberg’s advanced text on the subject (another standard reference for years to come), and proofs which appear in her book are only referred to in this one. There is a supplement consisting of corrections and giving solutions to the (many) problems listed in the book as open which have since been solved, largely by Shelah and by Doug Peterson. One assumes the supplement will be sent with the book, but if it is not it should be obtained from the author.
Reviewer: J.Roitman

MSC:

06E05 Structure theory of Boolean algebras
03E05 Other combinatorial set theory
54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets)
06-02 Research exposition (monographs, survey articles) pertaining to ordered structures
03-02 Research exposition (monographs, survey articles) pertaining to mathematical logic and foundations

Citations:

Zbl 0671.06001
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