Products of regular cardinals and cardinal invariants of products of Boolean algebras

1990 ◽  
Vol 70 (2) ◽  
pp. 129-187 ◽  
Author(s):  
Saharon Shelah
Keyword(s):  
Boolean Algebras ◽  
Cardinal Invariants ◽  
Regular Cardinals

scholarly journals TOWERS IN FILTERS, CARDINAL INVARIANTS, AND LUZIN TYPE FAMILIES

2018 ◽  
Vol 83 (3) ◽  
pp. 1013-1062 ◽  
Author(s):  
JÖRG BRENDLE ◽  
BARNABÁS FARKAS ◽  
JONATHAN VERNER
Keyword(s):  
List Type ◽  
Type Number ◽  
Cardinal Invariants ◽  
Regular Cardinals ◽  
Image Position

AbstractWe investigate which filters onωcan contain towers, that is, a modulo finite descending sequence without any pseudointersection (in${[\omega ]^\omega }$). We prove the following results:(1)Many classical examples of nice tall filters contain no towers (in ZFC).(2)It is consistent that tall analytic P-filters contain towers of arbitrary regular height (simultaneously for many regular cardinals as well).(3)It is consistent that all towers generate nonmeager filters (this answers a question of P. Borodulin-Nadzieja and D. Chodounský), in particular (consistently) Borel filters do not contain towers.(4)The statement “Every ultrafilter contains towers.” is independent of ZFC (this improves an older result of K. Kunen, J. van Mill, and C. F. Mills).Furthermore, we study many possible logical (non)implications between the existence of towers in filters, inequalities between cardinal invariants of filters (${\rm{ad}}{{\rm{d}}^{\rm{*}}}\left( {\cal F} \right)$,${\rm{co}}{{\rm{f}}^{\rm{*}}}\left( {\cal F} \right)$,${\rm{no}}{{\rm{n}}^{\rm{*}}}\left( {\cal F} \right)$, and${\rm{co}}{{\rm{v}}^{\rm{*}}}\left( {\cal F} \right)$), and the existence of Luzin type families (of size$\ge {\omega _2}$), that is, if${\cal F}$is a filter then${\cal X} \subseteq {[\omega ]^\omega }$is an${\cal F}$-Luzin family if$\left\{ {X \in {\cal X}:|X \setminus F| = \omega } \right\}$is countable for every$F \in {\cal F}$.

scholarly journals Densities of Ultraproducts of Boolean Algebras

1995 ◽  
Vol 47 (1) ◽  
pp. 132-145
Author(s):  
Sabine Koppelberg ◽  
Saharon Shelah
Keyword(s):  
Boolean Algebra ◽  
Boolean Algebras ◽  
Cardinal Invariants ◽  
The Third ◽  
Topological Density

AbstractWe answer three problems by J. D. Monk on cardinal invariants of Boolean algebras. Two of these are whether taking the algebraic density πA resp. the topological density cL4 of a Boolean algebra A commutes with formation of ultraproducts; the third one compares the number of endomorphisms and of ideals of a Boolean algebra.

scholarly journals Constructing Boolean Algebras for cardinal invariants

2001 ◽  
Vol 45 (4) ◽  
pp. 353-373
Author(s):  
Saharon Shelah
Keyword(s):  
Boolean Algebras ◽  
Cardinal Invariants

More on Cichoń's diagram and infinite games

2000 ◽  
Vol 65 (4) ◽  
pp. 1713-1724 ◽  
Author(s):  
Masaru Kada
Keyword(s):  
Boolean Algebras ◽  
Infinite Games ◽  
Cardinal Invariants ◽  
Cichoń’S Diagram

AbstractSome cardinal invariants from Cichoń's diagram can be characterized using the notion of cut-and-choose games on cardinals. In this paper we give another way to characterize those cardinals in terms of infinite games. We also show that some properties for forcing, such as the Sacks Property, the Laver Property and ωω-boundingness, are characterized by cut-and-choose games on complete Boolean algebras.

Applications of PCF theory

2000 ◽  
Vol 65 (4) ◽  
pp. 1624-1674 ◽  
Author(s):  
Saharon Shelah
Keyword(s):  
Independent Sets ◽  
Boolean Algebras ◽  
Pcf Theory ◽  
Cardinal Invariants

AbstractWe deal with several pcf problems: we characterize another version of exponentiation: maximal number of k-branches in a tree with λ nodes, deal with existence of independent sets in stable theories, possible cardinalities of ultraproducts and the depth of ultraproducts of Boolean Algebras. Also we give cardinal invariants for each λ with a pcf restriction and investigate further TD{f). The sections can be read independently, although there are some minor dependencies.

scholarly journals More on cardinal invariants of Boolean algebras

2000 ◽  
Vol 103 (1-3) ◽  
pp. 1-37 ◽  
Author(s):  
Andrzej Rosłanowski ◽  
Saharon Shelah
Keyword(s):  
Boolean Algebras ◽  
Cardinal Invariants

Geometric Cardinal Invariants, Maximal Functions and a Measure Theoretic Pigeonhole Principle

2005 ◽  
Vol 11 (4) ◽  
pp. 517-525
Author(s):  
Juris Steprāns
Keyword(s):  
Harmonic Analysis ◽  
Set Theory ◽  
Harmonic Functions ◽  
Maximal Functions ◽  
Pigeonhole Principle ◽  
Cardinal Invariants ◽  
Geometric Objects

AbstractIt is shown to be consistent with set theory that every set of reals of size ℵ1 is null yet there are ℵ1 planes in Euclidean 3-space whose union is not null. Similar results will be obtained for other geometric objects. The proof relies on results from harmonic analysis about the boundedness of certain harmonic functions and a measure theoretic pigeonhole principle.

CARDINAL INVARIANTS AND SETS OF REALS

1995 ◽  
Vol 21 (1) ◽  
pp. 78
Author(s):  
Bartoszyński
Keyword(s):  
Cardinal Invariants
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