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 to 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 97 are formulated.
Based on Cardinal Functions on Boolean Algebras (1990) by the same author, the present work is nearly twice the size of the original work. It contains solutions to many of the open problems which are discussed in greater detail than before. Among the new topics considered are ultraproducts and Fedorchuk?s theorem, and there is a more complete treatment of the cellularity of free products. Diagrams at the end of the book summarize the relationships between the functions for many important classes of Boolean algebras, including tree algebras and superatomic algebras.
Review:
"This book is an indispensable tool for anyone working in Boolean algebra, and is also recommended for set-theoretic topologists." - Zentralblatt MATH
Author(s): J. Donald Monk
Series: Progress in Mathematics
Edition: 1st ed. 1996. 2nd printing
Publisher: Birkhäuser Basel
Year: 2009
Language: English
Pages: 309
Cover......Page 1
Cardinal Invariants on Boolean Algebras......Page 4
Copyright - ISBN: 3034603339......Page 5
Foreword......Page 8
Table of contents
......Page 10
Definition of the cardinal functions considered.......Page 12
Some classifications of cardinal functions......Page 14
Algebraic properties of a single function.......Page 16
Derived operations.......Page 17
Other considerations......Page 18
Special classes of Boolean algebras......Page 19
Boolean products......Page 20
Boolean powers......Page 22
Set products......Page 26
One-point gluing......Page 27
The Aleksandroff duplicate......Page 29
The exponential......Page 30
Semigroup algebras......Page 36
Pseudo-tree algebras......Page 39
Simple extensions of Boolean algebras......Page 42
Minimal extensions of Boolean algebras......Page 44
Minimally generated Boolean algebras......Page 46
Tail algebras......Page 51
Initial chain algebras......Page 52
3. Cellularity......Page 56
4. Depth......Page 97
5. Topological density......Page 118
6. π-weight......Page 127
7. Length......Page 136
8. Irredundance......Page 144
9. Cardinality......Page 156
10. Independence......Page 158
11. π-Character......Page 165
12. Tightness......Page 175
13. Spread......Page 186
14. Character......Page 192
15. Hereditary Lindelöf degree......Page 201
16. Hereditary density......Page 207
17. Incomparability......Page 229
18. Hereditary cofinality......Page 237
19. Number of ultrafilters......Page 243
20. Number of automorphisms......Page 244
21. Number of endomorphisms......Page 247
22. Number of ideals......Page 249
23. Number of subalgebras......Page 250
Functions mentioned in the previous text......Page 255
Some additional natural functions......Page 256
Dimensions of Boolean algebras......Page 257
General case......Page 259
The main diagram, edges and “large” and “small” indications.......Page 260
The main diagram: no other relationships.......Page 261
The interval algebra diagram: the edges, indicated equalities, and the“large” and “small” indications. See below.
......Page 263
Interval algebras......Page 264
The tree algebra diagram: the indicated equalities and inequalities, and the “large” and “small” indications. See below.
......Page 265
Tree algebras......Page 266
The complete BA diagram: the indicated equalities and inequalities and the “large” and “small” indications. See below.
......Page 267
Diagram for superatomic BAs: the indicated relations, and the “large” and “small” indications. See below.
......Page 268
Superatomic BAs......Page 269
Superatomic BAs, no additional relationships:......Page 271
Atomic BAs......Page 272
Atomic diagram, edges and “large” and “small” indications.......Page 273
Atomic diagram, no other relations......Page 275
Atomless algebras.......Page 276
Semigroup algebras.......Page 277
Minimally generated BAs.......Page 279
Pseudo-tree algebras......Page 280
Minimally generated BAs......Page 281
3. The interval algebra on the reals.......Page 282
7. The Aleksandroff duplicate of a free BA.......Page 283
References......Page 290
Index of problems......Page 298
Index of symbols......Page 304
Index of names and words......Page 306
