Coherence of Semifilters
by T.Banakh and L.Zdomsky
The book is devoted to studying the (sub)coherence relation on semifilters, that is families of infinite
subsets of ω, closed under taking almost supersets.
On the family of ultrafilters the coherence relation was introduced in 80ies
by A.Blass who formulated his famous principle (NCF),
the Near Coherence of Filters, that found many nontrivial
applications in various
fields of mathematics.
In the book the (sub)coherence relation is
treated with help of cardinal functions defined on the lattice SF of semifilters.
Endowed with the Lawson topology the lattice SF becomes a supercompact
topological space. It can be interesting for
topological algebraists because any reasonable binary operation
on natural numbers induces a
righttopological operation on SF in the same way as it does on
the StoneCech compactification βω.
Contents
 Part 0:
Contents and Introduction (472Kb)
 Introduction
 Some history, warmup and motivation .. Introducing semifilters ..
The lattice SF of semifilters .. The limit operator on SF .. Algebraic operations on the lattice SF ..
(Sub)coherence relation ..
Near coherence of semifilters .. Characterizing meager and comeager semifilters ..
Strict subcoherence and regularity of semifilters .. The coherence lattice [SF] ..
Topologizing the coherence lattice [SF] .. Algebraic operations on [SF] ..
Cardinal characteristics of semifilters: general theory ..
Cardinal characteristics of semifilters: four levels of complexity ..
Cardinal characteristics of the first complexity level ..
The minimization π[] of the picharacter ..
The function representation of a semifilter ..
The nonification of the picharacter ..
``Ideal'' cardinal characteristics add[], cov[], non[], cof[] of semifilters ..
The relation ≤_{F} and its cardinal characteristics ..
Constructing noncoherent semifilters ..
Total coherence ..
The supremization of the linkedness number ..
Cardinal characteristics of rare semifilters ..
The structure of the coherence lattice [SF] .. Some Applications
 Part I:
Semifilters and duality (641Kb)
 Preliminaries
 Some standard notation and conventions ..
Some basic small cardinals .. Almost disjoint and independent families ..
Finitetoone functions.. Finitetofinite multifunctions
 Semifilters and Duality
 Semifilters ..
.. Duality ..
Linked and unsplit semifilters .. nLinked semifilters ..
Support of a semifilter .. Cardinal characteristics
of semifilters .. The Minimal Tower problem and cardinals t_{κ}
 Special Ultrafilters in βω
 The space of ultrafilters βω .. Ppoints and Qpoints ..
Weak Ppoints, OKpoints, Rpoints .. Dpoints
 Lattice SF of semifilters
 Lattices and their properties .. Symmetric lattices ..
Lattice SF of semifilters .. Limits by semifilters .. Algebraic operations on SF ..
Function representation of the lattice SF
 Part II:
Coherence Relation and Coherence Lattice (482Kb)
 Subcoherence preorder and its strict version
 (Sub)coherence of semifilters ..
Near coherent semifilters ..
Talagrand characterization of (co)meager semifilters ..
Strict (sub)coherence of semifilters ..
Regular semifilters
 Coherence Lattice [SF]
 Algebraic structure of [SF] .. The limit operator lim on [SF] ..
Topological properties of the (sub)coherence relation ..
Topologizing the coherence lattice [SF] .. Finite sublattices in [SF] ..
Compatibility of the algebraic structure and topology on [SF] .. Induced algebraic operations on [SF]
 Part III:
Cardinal characteristics of semifilters (844 Kb)
 Cardinal functions on the coherence lattice: General theory

Various properties of cardinal characteristics ..
Continuity of cardinal characteristics on [SF] ..
A general result on total coherence ..
ξminimal and ξmaximal semifilters ..
A Maximal Linked Dichotomy ..
(NCF) and other Coherence Principles
 The πcharacter π(F) of a semifilter
 Minimization π[] of the πcharacter ..
Algebraic and continuity properties of πχ[] and ad[] ..
Calculating the cardinals π_{b} and π_{f} ..
Calculating the cardinals π_{l} and π_{u} ..
The function representation of a semifilter ..
The nonification of the πcharacter ..
Interplay between πχ[] and ad[] ..
Small cardinals r_{κ} and u_{κ}
 Cardinal characteristics of semifilters having ``ideal''
origin
 Ideals and their cardinal characteristics ..
``Ideal'' cardinal characteristics of families of semifilters ..
Calculating the covering number for various families of semifilters ..
``Ideal'' cardinal characteristics of semifilters
 (Co)boundedness numbers b(F) and q(F) of a semifilter
 The relation ≤_{F} and its cardinal characteristics ..
Oriented subcoherence and interplay between b(F) and q(F) ..
Interplay between b(), q() and πχ[] ..
(Co)boundedness number of a family of semifilters ..
Cardinal characteristics of a semifilter and its support ..
Interplay between cardinal characteristics of a semifilter ..
Detecting the coherence of ultrafilters by monotone surjections
 Part IV:
Noncoherent and coherent semifilters (520Kb)
 Constructing noncoherent semifilters
 Noncoherent semifilters with
large almost disjointness number ..
Constructing noncoherent (ultra)filters ..
Constructing noncoherent nlinked semifilters ..
Constructing 2^{r} incomparable semifilters with centered union ..
Embedding SF into [SF] under (r≥d) ..
Totally separated sequences of ultrafilters under (u≠d)
 Total coherence
 Coherence of Simon semifilters under
bb .. πχMinimal semifilters under (r<d) ..
NCF and other coherence principles
 The linkedness number πp[F] of a semifilter

Interplay between πp[] and ``ideal'' cardinal characteristics ..
The small cardinals πp^{u} and πp^{u} ..
πpmaximal semifilters under b<πp^{u} ..
The interplay between πχ[] and πp[] ..
Some implications of (r<s) ..
Applications to simple Ppoints ..
Cardinal characteristics of rare semifilters ..
Cardinal characteristics under (r<g) or (r<s)
 Part V:
Applications and Open Problems (619 Kb)
 Consistency Properties of the Coherence Lattice [SF]
 Structure of the space [UF] ..
Isolated points of [SF] ..
Sublattices of [SF] and their cardinal characteristics ..
The size of [SF] and some of its subsets .. Completeness of the coherence lattice ..
 Applications
 The structure of the StoneCech
remainder of the halfline ..
Additivity of the Menger property ..
Discontinuous separately continuous functions ..
Some Open Problems
 References
This is a preliminary version of the book. So the authors
would appreciate any remarks, comments, and suggestions.
You can write to the authors by email tbanakh@yahoo.com.