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Birkhäuser - Birkhäuser Mathematics | Cardinal Invariants on Boolean Algebras - Second Revised Edition

Cardinal Invariants on Boolean Algebras

Second Revised Edition

Series: Progress in Mathematics, Vol. 142

Monk, J. Donald

Originally published as volume 142 in the series: Progress in Mathematics

2nd ed. 2014, VII, 573 p. 15 illus.

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  • 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

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.

Content Level » Research

Keywords » Boolean algebra - cardinal functions - cellularity - independence - reaping number

Related subjects » Birkhäuser Mathematics

Table of contents 

Introduction.- 1. Special Operations on Boolean Algebras.- 2. Special Classes of Boolean Algebras.- 3. Cellularity.- 4. Depth.- 5. Topological Density.- 6. Pi-Weight.- 7. Length.- 8. Irredundance.- 9. Cardinality.- 10. Independence.- 11. Pi-Character.- 12. Tightness.- 13. Spread.- 14. Character.- 15. Hereditary Lindelöf Degree.- 16. Hereditary Density.- 17. Incomparability.- 18. Hereditary Cofinality.- 19. Number of Ultrafilters.- 20. Number of Automorphisms.- 21. Number of Endomorphisms.- 22. Number of Ideals.- 23. Number of Subalgebras.- 24. Other Cardinal Functions.- 25. Diagrams.- 26. Examples.- 27. Problems.- References.- Symbol Index.- Subject Index.- Name Index.

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