Ebook: Cardinal Invariants on Boolean Algebras

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

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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

Content:
Front Matter....Pages i-ix
Introduction....Pages 1-8
Special operations on Boolean algebras....Pages 9-24
Special classes of Boolean algebras....Pages 25-44
Cellularity....Pages 45-85
Depth....Pages 86-106
Topological density....Pages 107-115
?-weight....Pages 116-124
Length....Pages 125-132
Irredundance....Pages 133-144
Cardinality....Pages 145-146
Independence....Pages 147-153
?-Character....Pages 154-163
Tightness....Pages 164-174
Spread....Pages 175-180
Character....Pages 181-189
Hereditary Lindel?f degree....Pages 190-195
Hereditary density....Pages 196-217
Incomparability....Pages 218-225
Hereditary cofinality....Pages 226-231
Number of ultrafilters....Pages 232-232
Number of automorphisms....Pages 233-235
Number of endomorphisms....Pages 236-237
Number of ideals....Pages 238-238
Number of subalgebras....Pages 239-243
Other cardinal functions....Pages 244-247
Diagrams....Pages 248-270
Examples....Pages 271-277
Back Matter....Pages 279-298

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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

Content:
Front Matter....Pages i-ix
Introduction....Pages 1-8
Special operations on Boolean algebras....Pages 9-24
Special classes of Boolean algebras....Pages 25-44
Cellularity....Pages 45-85
Depth....Pages 86-106
Topological density....Pages 107-115
?-weight....Pages 116-124
Length....Pages 125-132
Irredundance....Pages 133-144
Cardinality....Pages 145-146
Independence....Pages 147-153
?-Character....Pages 154-163
Tightness....Pages 164-174
Spread....Pages 175-180
Character....Pages 181-189
Hereditary Lindel?f degree....Pages 190-195
Hereditary density....Pages 196-217
Incomparability....Pages 218-225
Hereditary cofinality....Pages 226-231
Number of ultrafilters....Pages 232-232
Number of automorphisms....Pages 233-235
Number of endomorphisms....Pages 236-237
Number of ideals....Pages 238-238
Number of subalgebras....Pages 239-243
Other cardinal functions....Pages 244-247
Diagrams....Pages 248-270
Examples....Pages 271-277
Back Matter....Pages 279-298
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