Overview
- Covers most of what is known on cardinal invariants in Boolean algebras
- Includes a list of 185 interesting open problems
- Detailed study of 21 cardinal number valued functions defined for any Boolean algebra
Part of the book series: Progress in Mathematics (PM, volume 142)
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Table of contents (28 chapters)
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Front Matter
Keywords
About this book
This book is concerned with cardinal number valued functions defined for any Boolean algebra. Examples of such functions are independence, which assigns to each Boolean algebra the supremum of the cardinalities of its free subalgebras, and cellularity, which gives the supremum of cardinalities of sets of pairwise disjoint elements. Twenty-one such functions are studied in detail, and many more in passing. The questions considered are the behaviour of these functions under algebraic operations such as products, free products, ultraproducts, and their relationships to one another.
Assuming familiarity with only the basics of Boolean algebras and set theory, through simple infinite combinatorics and forcing, the book reviews current knowledge about these functions, giving complete proofs for most facts. A special feature of the book is the attention given to open problems, of which 185 are formulated.
Based on Cardinal Functions on Boolean Algebras (1990) and Cardinal Invariants on Boolean Algebras (1996) by the same author, the present work is much larger than either of these. It contains solutions to many of the open problems of the earlier volumes. Among the new topics are continuum cardinals on Boolean algebras, with a lengthy treatment of the reaping number. Diagrams at the end of the book summarize the relationships between the functions for many important classes of Boolean algebras, including interval algebras, tree algebras and superatomic algebras.
Reviews
“This book surveys the theory of cardinal-valued functions for Boolean algebras … . The book provides an invaluable resource for anyone working in Boolean algebras and can also be interesting for people working in set-theoretic topology or combinatorial set theory.” (Juan Carlos MartĂnez, Mathematical Reviews, April, 2015)
Authors and Affiliations
Bibliographic Information
Book Title: Cardinal Invariants on Boolean Algebras
Book Subtitle: Second Revised Edition
Authors: J. Donald Monk
Series Title: Progress in Mathematics
DOI: https://doi.org/10.1007/978-3-0348-0730-2
Publisher: Birkhäuser Basel
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer Basel 2014
Hardcover ISBN: 978-3-0348-0729-6Published: 21 February 2014
eBook ISBN: 978-3-0348-0730-2Published: 11 February 2014
Series ISSN: 0743-1643
Series E-ISSN: 2296-505X
Edition Number: 2
Number of Pages: VII, 573
Number of Illustrations: 15 b/w illustrations
Additional Information: Originally published as volume 142 in the series: Progress in Mathematics
Topics: Mathematical Logic and Foundations, Order, Lattices, Ordered Algebraic Structures