Cardinal Invariants on Boolean Algebras : Second Revised Edition / by J. Donald Monk.

Format:

Book

Author/Creator:

Monk, J. Donald., Author.

Series:

Progress in Mathematics, 0743-1643 ; 142

Language:

English

Subjects (All):

Logic, Symbolic and mathematical.

Algebra.

Ordered algebraic structures.

Mathematical Logic and Foundations.

Order, Lattices, Ordered Algebraic Structures.

Local Subjects:

Mathematical Logic and Foundations.

Order, Lattices, Ordered Algebraic Structures.

Physical Description:

1 online resource (569 pages) : illustrations.

Edition:

2nd ed. 2014.

Place of Publication:

Basel : Springer Basel : Imprint: Birkhäuser, 2014.

Language Note:

English

Summary:

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.

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.

Notes:

Bibliographic Level Mode of Issuance: Monograph

Includes bibliographical references and indexes.

ISBN:

3-0348-0730-9