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 80-ies
by A.Blass who formulated his famous principle (NCF),
the Near Coherence of Filters, that found many non-trivial
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
right-topological operation on SF in the same way as it does on
the Stone-Cech compactification βω.
Contents
- Part 0:
Contents and Introduction (472Kb)
- Introduction
- Some history, warm-up 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 pi-character ..
The function representation of a semifilter ..
The nonification of the pi-character ..
``Ideal'' cardinal characteristics add[-], cov[-], non[-], cof[-] of semifilters ..
The relation ≤F and its cardinal characteristics ..
Constructing non-coherent 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 ..
Finite-to-one functions.. Finite-to-finite multifunctions
Semifilters and Duality
- Semifilters ..
.. Duality ..
Linked and unsplit semifilters .. n-Linked semifilters ..
Support of a semifilter .. Cardinal characteristics
of semifilters .. The Minimal Tower problem and cardinals tκ
Special Ultrafilters in βω
- The space of ultrafilters βω .. P-points and Q-points ..
Weak P-points, OK-points, R-points .. D-points
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 ..
ξ-minimal and ξ-maximal semifilters ..
A Maximal Linked Dichotomy ..
(NCF) and other Coherence Principles
The π-character π(F) of a semifilter
- Minimization π[-] of the π-character ..
Algebraic and continuity properties of πχ[-] and ad[-] ..
Calculating the cardinals πb and πf ..
Calculating the cardinals πl and πu ..
The function representation of a semifilter ..
The nonification of the π-character ..
Interplay between πχ[-] and ad[-] ..
Small cardinals rκ and uκ
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:
Non-coherent and coherent semifilters (520Kb)
Constructing non-coherent semifilters
- Non-coherent semifilters with
large almost disjointness number ..
Constructing noncoherent (ultra)filters ..
Constructing non-coherent n-linked semifilters ..
Constructing 2r 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 .. πχ-Minimal 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 πpu and πpu ..
πp-maximal semifilters under b<πpu ..
The interplay between πχ[-] and πp[-] ..
Some implications of (r<s) ..
Applications to simple P-points ..
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 Stone-Cech
remainder of the half-line ..
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 e-mail tbanakh@yahoo.com.
