Cardinal Invariants on Boolean Algebras

This text covers cardinal number valued functions defined for any Boolean algebra such as cellularity. It explores the behavior of these functions under algebraic operations such as products, free products, ultraproducts and their relationships to each other.

This is a greatly revised and expanded version of the book Cardinal fu- tionsonBooleanalgebras,Birkh auser1990.Knownmistakesinthatbookhave beencorrected,andmanyoftheproblemsstatedtherehavesolutionsinthepresent treatment. At the same time, many new problems are formulated here; some as development of the solved problems from the earlier work, but most as a result of more careful study of the notions. The book is supposed to be self-contained, and for that reason many classical results are included. ForhelponthisbookIwishtothankE.K.vanDouwen,K.Grant,L. Heindorf, I. Juh asz, S. Koppelberg, P. Koszmider, P. Nyikos, D. Peterson, M. Rubin, S. Shelah, and S. Todor? cevi c. Unpublished results of some of these people are contained here, sometimes with proofs, with their permission. As the reader will see, my greatest debt is to Saharon Shelah, who has worked on, and solved, many of the problems stated in the 1990 book as well as in preliminary versions of this book. Of course I am always eager to hear about solutions of problems, mistakes, etc. Electronic lists of errata and the status of the open problems are maintained, initially on the anonymous ftp server of euclid.colorado.edu, directory pub/babib; on www, go to ftp://euclid.colorado.edu/pub/babib.

Covers most of what was known on cardinal invariants in Boolean algebras List of 97 interesting open problems Includes supplementary material: sn.pub/extras

Klappentext

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. 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 Fedorchukís 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. "This book is an indispensable tool for anyone working in Boolean algebra, and is also recommended for set-theoretic topologists." Zentralblatt MATH TOC:Introduction.- Special operations on Boolean algebras.- Special classes of Boolean algebras.- Cellularity.- Depth.- Topological density.- pi-weight.- Length.- Irredundance.- Cardinality.- Independence.- pi-character.- Tightness.- Spread.- Character.- Hereditary Lindelöf degree.- Hereditary density.- Incomparability.- Hereditary cofinality.- Number of ultrafilters, automorphisms, endomorphisms, ideals, subalgebras.- Other cardinal functions.- Diagrams.- Examples.

Inhalt
Special operations on Boolean algebras.- Special classes of Boolean algebras.- Cellularity.- Depth.- Topological density.- ?-weight.- Length.- Irredundance.- Cardinality.- Independence.- ?-Character.- Tightness.- Spread.- Character.- Hereditary Lindelöf degree.- Hereditary density.- Incomparability.- Hereditary cofinality.- Number of ultrafilters.- Number of automorphisms.- Number of endomorphisms.- Number of ideals.- Number of subalgebras.- Other cardinal functions.- Diagrams.- Examples.