scholarly journals Some Applications of the Katětov Order on Borel Ideals

2016 ◽  
pp. 1-8
Author(s):  
Nikodem Mrożek
Keyword(s):  
Katětov Order ◽  
Borel Ideals

On the structure of Borel ideals in-between the ideals ED and Fin ⊗ Fin in the Katětov order

2021 ◽  
pp. 102976
Author(s):  
Pratulananda Das ◽  
Rafał Filipów ◽  
Szymon Gła̧b ◽  
Jacek Tryba
Keyword(s):  
Katětov Order ◽  
Borel Ideals

Katětov order on Borel ideals

2017 ◽  
Vol 56 (7-8) ◽  
pp. 831-847 ◽  
Author(s):  
Michael Hrušák
Keyword(s):  
Katětov Order ◽  
Borel Ideals

Some Structural Aspects of the Katětov Order on Borel Ideals

Order ◽  
2015 ◽  
Vol 33 (2) ◽  
pp. 189-194 ◽  
Author(s):  
Osvaldo Guzmán-González ◽  
David Meza-Alcántara
Keyword(s):  
Katětov Order ◽  
Borel Ideals ◽  
Structural Aspects

scholarly journals The Ramsey property implies no mad families

2019 ◽  
Vol 116 (38) ◽  
pp. 18883-18887 ◽  
Author(s):  
David Schrittesser ◽  
Asger Törnquist
Keyword(s):  
Set Theory ◽  
Large Set ◽  
Descriptive Set Theory ◽  
Ramsey Property ◽  
Mad Families ◽  
Almost Disjoint ◽  
Descriptive Set ◽  
Maximal Almost Disjoint ◽  
Borel Ideals ◽  
Standing Problem

We show that if all collections of infinite subsets of N have the Ramsey property, then there are no infinite maximal almost disjoint (mad) families. The implication is proved in Zermelo–Fraenkel set theory with only weak choice principles. This gives a positive solution to a long-standing problem that goes back to Mathias [A. R. D. Mathias, Ann. Math. Logic 12, 59–111 (1977)]. The proof exploits an idea which has its natural roots in ergodic theory, topological dynamics, and invariant descriptive set theory: We use that a certain function associated to a purported mad family is invariant under the equivalence relation E0 and thus is constant on a “large” set. Furthermore, we announce a number of additional results about mad families relative to more complicated Borel ideals.

scholarly journals Rigidity properties of Borel ideals on the integers

1998 ◽  
Vol 85 (1-3) ◽  
pp. 195-205 ◽  
Author(s):  
Alexander S. Kechris
Keyword(s):  
Borel Ideals

scholarly journals Borel degenerations of arithmetically Cohen–Macaulay curves in ℙ3

2014 ◽  
Vol 24 (05) ◽  
pp. 715-739 ◽  
Author(s):  
Gunnar Fløystad ◽  
Margherita Roggero
Keyword(s):  
Hilbert Scheme ◽  
Codimension Two ◽  
Borel Ideal ◽  
Borel Ideals

We investigate Borel ideals on the Hilbert scheme components of arithmetically Cohen–Macaulay (ACM) codimension two schemes in ℙn. We give a basic necessary criterion for a Borel ideal to be on such a component. Then considering ACM curves in ℙ3 on a quadric we compute in several examples all the Borel ideals on their Hilbert scheme component. Based on this we conjecture which Borel ideals are on such a component, and for a range of Borel ideals we prove that they are on the component.

Continuum-many Boolean algebras of the form Borel

2004 ◽  
Vol 69 (3) ◽  
pp. 799-816 ◽  
Author(s):  
Michael Ray Oliver
Keyword(s):  
Equivalence Relation ◽  
Continuum Hypothesis ◽  
Boolean Algebras ◽  
New Technique ◽  
Finite Sets ◽  
A New Technique ◽  
Borel Ideals ◽  
Forcing Axiom ◽  
The Continuum ◽  
The Ideal

Abstract.We examine the question of how many Boolean algebras, distinct up to isomorphism, that are quotients of the powerset of the naturals by Borel ideals, can be proved to exist in ZFC alone. The maximum possible value is easily seen to be the cardinality of the continuum ; earlier work by Ilijas Farah had shown that this was the value in models of Martin's Maximum or some similar forcing axiom, but it was open whether there could be fewer in models of the Continuum Hypothesis.We develop and apply a new technique for constructing many ideals whose quotients must be nonisomorphic in any model of ZFC. The technique depends on isolating a kind of ideal, called shallow, that can be distinguished from the ideal of all finite sets even after any isomorphic embedding, and then piecing together various copies of the ideal of all finite sets using distinct shallow ideals. In this way we are able to demonstrate that there are continuum-many distinct quotients by Borel ideals, indeed by analytic P-ideals, and in fact that there is in an appropriate sense a Borel embedding of the Vitali equivalence relation into the equivalence relation of isomorphism of quotients by analytic P-ideals. We also show that there is an uncountable definable wellordered collection of Borel ideals with distinct quotients.

scholarly journals Fréchet Borel ideals with Borel orthogonal

2018 ◽  
Vol 152 (1) ◽  
pp. 141-163 ◽  
Author(s):  
Francisco Guevara ◽  
Carlos Uzcátegui
Keyword(s):  
Borel Ideals

When does the Katětov order imply that one ideal extends the other?

2013 ◽  
Vol 130 (1) ◽  
pp. 91-102 ◽  
Author(s):  
Paweł Barbarski ◽  
Rafał Filipów ◽  
Nikodem Mrożek ◽  
Piotr Szuca
Keyword(s):  
The Other ◽  
Katětov Order
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