© 2014

Cardinal Invariants on Boolean Algebras

Second Revised Edition


Part of the Progress in Mathematics book series (PM, volume 142)

Table of contents

  1. Front Matter
    Pages i-vii
  2. J. Donald Monk
    Pages 1-9
  3. J. Donald Monk
    Pages 11-34
  4. J. Donald Monk
    Pages 35-82
  5. J. Donald Monk
    Pages 83-154
  6. J. Donald Monk
    Pages 155-218
  7. J. Donald Monk
    Pages 219-235
  8. J. Donald Monk
    Pages 237-277
  9. J. Donald Monk
    Pages 279-290
  10. J. Donald Monk
    Pages 291-310
  11. J. Donald Monk
    Pages 311-334
  12. J. Donald Monk
    Pages 335-372
  13. J. Donald Monk
    Pages 373-385
  14. J. Donald Monk
    Pages 387-404
  15. J. Donald Monk
    Pages 405-420
  16. J. Donald Monk
    Pages 421-435
  17. J. Donald Monk
    Pages 437-448
  18. J. Donald Monk
    Pages 449-460
  19. J. Donald Monk
    Pages 461-480
  20. J. Donald Monk
    Pages 481-488

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.


Boolean algebra cardinal functions cellularity independence reaping number

Authors and affiliations

  1. 1.Department of MathematicsUniversity of ColoradoBoulderUSA

Bibliographic information

  • Book Title Cardinal Invariants on Boolean Algebras
  • Book Subtitle Second Revised Edition
  • Authors J. Donald Monk
  • Series Title Progress in Mathematics
  • Series Abbreviated Title Progress in Mathematics(Birkhäuser)
  • DOI
  • Copyright Information Springer Basel 2014
  • Publisher Name Birkhäuser, Basel
  • eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
  • Hardcover ISBN 978-3-0348-0729-6
  • eBook ISBN 978-3-0348-0730-2
  • Series ISSN 0743-1643
  • Series E-ISSN 2296-505X
  • Edition Number 2
  • Number of Pages VII, 573
  • Number of Illustrations 15 b/w illustrations, 0 illustrations in colour
  • Additional Information Originally published as volume 142 in the series: Progress in Mathematics
  • Topics Mathematical Logic and Foundations
    Order, Lattices, Ordered Algebraic Structures
  • Buy this book on publisher's site
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“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 Martínez, Mathematical Reviews, April, 2015)