Table Of ContentProgress in Mathematics
142
J. Donald Monk
Cardinal
Invariants on
Boolean Algebras
Second Revised Edition
Progress in Mathematics
Volume 142
Series Editors
Hyman Bass, University of Michigan, Ann Arbor, USA
Jiang-Hua Lu, The University of Hong Kong, Hong Kong SAR, China
Joseph Oesterlé, Université Pierre et Marie Curie, Paris, France
Yuri Tschinkel, Courant Institute of Mathematical Sciences, New York, USA
For further volumes:
http://www.springer.com/series/4848
J. Donald Monk
Cardinal Invariants
on Boolean Algebras
Second Revised Edition
ISSN 0743-1643 ISSN 2296-505X (electronic)
ISBN 978-3-0348-0729-6 ISBN 978-3-0348-0730-2 (eBook)
DOI10.1007/978-3-0348-0730-2
Springer Basel Heidelberg New York Dordrecht London
Library of Congress Control Number: 2014932538
This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the
material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation,
broadcasting, reproduction on microfilms or in any other physical way, and transmission or information
storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now
known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with
reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed
on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts
thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current
version, and permission for use must always be obtained from Springer. Permissions for use may be obtained
through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the
respective Copyright Law.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication
does not imply, even in the absence of a specific statement, that such names are exempt from the relevant
protective laws and regulations and therefore free for general use.
While the advice and information in this book are believed to be true and accurate at the date of publication,
neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or
omissions that may be made. The publisher makes no warranty, express or implied, with respect to the
material contained herein.
Printed on acid-free paper
Springer Basel is part of Springer Science+Business Media (www.birkhauser-science.com)
Contents
Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
0 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1 Special Operations on Boolean Algebras . . . . . . . . . . . . . . . . 11
2 Special Classes of Boolean Algebras . . . . . . . . . . . . . . . . . . 35
3 Cellularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4 Depth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
5 Topological Density . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
6 π-Weight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
7 Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
8 Irredundance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
9 Cardinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311
10 Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335
11 π-Character . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373
12 Tightness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387
13 Spread . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405
14 Character . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421
15 Hereditary Lindel¨of Degree. . . . . . . . . . . . . . . . . . . . . . . 437
16 Hereditary Density . . . . . . . . . . . . . . . . . . . . . . . . . . . 449
17 Incomparability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461
18 Hereditary Cofinality . . . . . . . . . . . . . . . . . . . . . . . . . . 481
19 Number of Ultrafilters . . . . . . . . . . . . . . . . . . . . . . . . . 489
v
vi Contents
20 Number of Automorphisms . . . . . . . . . . . . . . . . . . . . . . . 491
21 Number of Endomorphisms. . . . . . . . . . . . . . . . . . . . . . . 495
22 Number of Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497
23 Number of Subalgebras . . . . . . . . . . . . . . . . . . . . . . . . . 499
24 Other Cardinal Functions . . . . . . . . . . . . . . . . . . . . . . . . 505
25 Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509
26 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531
27 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 551
Symbol Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561
Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 567
Name Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 571
Foreword
This book is the successorof Cardinal functions on Boolean algebras (Birkh¨auser
1990)and Cardinal invariants on Boolean algebras (Birkh¨auser1996). It contains
most of the material of these books, and adds the following:
(1) Indication of the progress made on the open problems formulated in the
earlier versions, with detailed solutions in many cases.
(2) Inclusion of some new cardinal functions, mainly those associated with con-
tinuum cardinals.
Thematerialonsheaves,Booleanproducts,andBooleanpowershasbeenomitted,
since these no longer play a role in our discussion of the cardinal invariants.
Although many problems in the earlier versions have been solved, many of
them are still open. In this edition we repeat those unsolved problems, and add
severalnew ones.
J. Donald Monk
Boulder, Colorado
[email protected]
vii
0 Introduction
Thisbookisconcernedwiththetheoryofcertainnaturalfunctionsk whichassign
toeachinfiniteBooleanalgebraAacardinalnumberk(A)orasetk(A)ofcardinal
numbers.ThepurposeofthebookistosurveythisareaofthetheoryofBAs,giving
proofs for a large number of results, some of which are new, mentioning most of
theknownresults,andformulatingopenproblems.Someoftheopenproblemsare
somewhat vague (“Characterize...” or something like that), but frequently these
areevenmoreimportant thanthe specific problems we state; sowe have optedto
enumerate problems of both sorts in order to focus attention on them.
The framework that we shall set forth and then follow in investigating car-
dinal functions seems to us to be important for several reasons. First of all, the
functions themselves seem intrinsically interesting. Many of the questions which
naturally arise can be easily answered on the basis of our current knowledge of
the structure of Boolean algebras, but some of these answers require rather deep
arguments of set theory, algebra, or topology. This provides another interest in
their study: as a natural source of applications of set-theoretical, algebraic, or
topological methods. Some of the unresolved questions are rather obscure and
uninteresting, but some of them have a general interest. Altogether, the study of
cardinalfunctionsseemstobringaunityanddepthtomanyisolatedinvestigations
in the theory of BAs.
ThereareseveralsurveysofcardinalfunctionsonBooleanalgebras,or,more
generally, on topological spaces: See Arhangelski˘ı [78], Comfort [71], van Douwen
[89], Hodel [84], Juha´sz [75], Juha´sz [80], Juha´sz [84], Monk [84], Monk [90], and
Monk[96].We shallnotassumeanyacquaintancewithanyofthese. Onthe other
hand, we shall frequently refer to results proved in Part I of the Handbook of
Boolean(cid:2)Algebras, Koppelberg [89]. One additional bit of terminology: in a weak
product w A , we call an element a of type 1 iff {i∈I :a (cid:3)=0}, the 1-support
i∈I i i
of a, is finite, and of type 2 iff {i∈I :a (cid:3)=1}, called the 2-support of a, is finite.
i
We have not attempted to give a complete history of the results mentioned
in this book. The references can be consulted for a detailed background.
J.D. Monk, Cardinal Invariants on Boolean Algebras: Second Revised Edition, 1
Progress in Mathematics 142, DOI 10.1007/978-3-0348-0730-2_1,
© Springer Basel 2014
2 Chapter 0. Introduction
Definition of the main cardinal functions considered
Wedeferuntillaterthediscussionoftheexistenceofsomeofthesefunctions;they
do not all exist for every BA.
Cellularity. A subset X of a BA A is called disjoint if its members are pairwise
disjoint. The cellularity of A, denoted by c(A), is
sup{|X|:X is a disjoint subset of A}.
Depth. Depth(A) is
sup{|X|:X is a subset of A well ordered by the Boolean ordering}.
Topological density. The density of a topological space X, denoted by d(X), is
the smallest cardinal κ such that X has a dense subspace of cardinality κ. The
topological density of a BA A, also denoted by d(A), is the density of its Stone
space Ult(A).
π-weight.AsubsetX ofaBAAisdenseinAifforalla∈A+ thereisanx∈X+
such that x≤a. The π-weight of a BA A, denoted by π(A), is
min{|X|:X is dense in A}.
This couldalsobe calledthe algebraic density of A. (Recall thatfor any subsetX
of a BA, X+ is the collection of nonzero elements of X.)
Length. Length(A) is
sup{|X|:X is a chain in A}.
Irredundance.AsubsetX ofaBAAisirredundant ifforallx∈X,x∈/ (cid:5)X\{x}(cid:6).
(Recallthat(cid:5)Y(cid:6)isthesubalgebrageneratedbyY.)TheirredundanceofA,denoted
by Irr(A), is
sup{|X|:X is an irredundant subset of A}.
Cardinality. This is just |A|. Sometimes we denote it by card(A).
Independence.AsubsetX ofAiscalledindependentifX isasetoffreegenerators
for (cid:5)X(cid:6). Then the independence of A, denoted by Ind(A), is
sup{|X|:X is an independent subset of A}.
π-character. For any ultrafilter F on A, let πχ(F)=min{|X|:X is dense in F}.
Note here that it is not requiredthat X ⊆F. Then the π-character of A, denoted
by πχ(A), is
sup{πχ(F):F is an ultrafilter of A}.
