Cardinal Invariants On Boolean Algebras

Front Cover
Springer Science & Business Media, 1996 - Mathematics - 298 pages

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 FedorchukA-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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Contents

I
1
II
9
III
25
IV
43
V
84
VI
104
VII
112
VIII
121
XVII
192
XVIII
214
XIX
222
XX
228
XXI
229
XXII
232
XXIII
234
XXIV
235

IX
129
X
141
XI
143
XII
150
XIII
160
XIV
171
XV
177
XVI
186
XXV
240
XXVI
244
XXVII
267
XXVIII
274
XXIX
282
XXX
288
XXXI
290
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