No minimal tall Borel ideal in the Katětov order

§1. Introduction. Ideals and filters of subsets of natural numbers have been studied by set theorists and topologists for a long time. There is a vast literature concerning various kinds of ultrafilters (or, dually, maximal ideals). There is also a substantial interest in nicely definable (Borel, analytic) ideals—these by old results of Sierpiński are very far from being maximal— and the structure of such ideals will concern us in this announcement. In addition to being interesting in their own right, Borel and analytic ideals occur naturally in the investigations of compact subsets of the space of all Baire class 1 functions on a Polish space (Rosenthal compacta), see [12, 18]. Also, certain objects associated with such ideals are of considerable interest and were quite extensively studied by several authors. Let us list here three examples; in all three of them I stands for an analytic or Borel ideal.1. The partial order induced by I on P(ω): X ≥I Y iff X \ Y ϵ I ([16]) and the partial order (I, ⊂)([18]).2. Boolean algebras of the form P(ω)/I and their automorphisms ([6, 5, 19, 20]).3. The equivalence relation associated with I: XEI Y iff X Δ ϵ I ([4, 14, 15,9]).In Section 4, we will have an opportunity to state some consequences of our results for equivalence relations as in 3.

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Borel degenerations of arithmetically Cohen–Macaulay curves in ℙ3

International Journal of Algebra and Computation ◽

10.1142/s0218196714500301 ◽

2014 ◽

Vol 24 (05) ◽

pp. 715-739 ◽

Cited By ~ 4

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.

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

Colloquium Mathematicum ◽

10.4064/cm130-1-9 ◽

2013 ◽

Vol 130 (1) ◽

pp. 91-102 ◽

Cited By ~ 4

Author(s):

Paweł Barbarski ◽

Rafał Filipów ◽

Nikodem Mrożek ◽

Piotr Szuca

Keyword(s):

The Other ◽

Katětov Order

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Katětov order on Borel ideals

Archive for Mathematical Logic ◽

10.1007/s00153-017-0543-x ◽

2017 ◽

Vol 56 (7-8) ◽

pp. 831-847 ◽

Cited By ~ 6

Author(s):

Michael Hrušák

Keyword(s):

Katětov Order ◽

Borel Ideals

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Borel complexity of isomorphism between quotient Boolean algebras

Journal of Symbolic Logic ◽

10.2178/jsl/1230396922 ◽

2008 ◽

Vol 73 (4) ◽

pp. 1328-1340

Author(s):

Su Gao ◽

Michael Ray Oliver

Keyword(s):

Equivalence Relation ◽

Boolean Algebras ◽

Equivalence Relations ◽

Borel Set ◽

Borel Sets ◽

Opposite Case ◽

Borel Complexity ◽

Borel Ideal ◽

The One ◽

Ergodic Equivalence Relations

In response to a question of Farah, “How many Boolean algebras are there?” [Far04], one of us (Oliver) proved that there are continuum-many nonisomorphic Boolean algebras of the form with I a Borel ideal on the natural numbers, and in fact that this result could be improved simultaneously in two directions:(i) “Borel ideal” may be improved to “analytic P-ideal”(ii) “continuum-many” may be improved to “E0-many”; that is, E0 is Borel reducible to the isomorphism relation on quotients by analytic P-ideals.See [Oli04].In [AdKechOO], Adams and Kechris showed that the relation of equality on Borel sets (and therefore, any Borel equivalence relation whatsoever) is Borel reducible to the equivalence relation of Borel bireducibility. (In somewhat finer terms, they showed that the partial order of inclusion on Borel sets is Borel reducible to the quasi-order of Borel reducibility.) Their technique was to find a collection of, in some sense, strongly mutually ergodic equivalence relations, indexed by reals, and then assign to each Borel set B a sort of “direct sum” of the equivalence relations corresponding to the reals in B. Then if B1, ⊆ B2 it was easy to see that the equivalence relation thus induced by B1 was Borel reducible to the one induced by B2, whereas in the opposite case, taking x to be some element of B / B2, it was possible to show that the equivalence relation corresponding to x, which was part of the equivalence relation induced by B1, was not Borel reducible to the equivalence relation corresponding to B2.

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Cofinal families of Borel equivalence relations and quasiorders

Journal of Symbolic Logic ◽

10.2178/jsl/1129642127 ◽

2005 ◽

Vol 70 (4) ◽

pp. 1325-1340 ◽

Cited By ~ 8

Author(s):

Christian Rosendal

Keyword(s):

Equivalence Relation ◽

Metric Spaces ◽

Equivalence Relations ◽

Borel Equivalence Relations ◽

Compact Metric Spaces ◽

Borel Equivalence Relation ◽

Borel Ideal ◽

Borel Reducibility

AbstractFamilies of Borel equivalence relations and quasiorders that are cofinal with respect to the Borel reducibility ordering. ≤B, are constructed. There is an analytic ideal on ω generating a complete analytic equivalence relation and any Borel equivalence relation reduces to one generated by a Borel ideal. Several Borel equivalence relations, among them Lipschitz isomorphism of compact metric spaces, are shown to be Kσ complete.

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Some applications of the theory of Katětov order to ideal convergence

Topology and its Applications ◽

10.1016/j.topol.2020.107545 ◽

2020 ◽

pp. 107545

Author(s):

Hang Zhang ◽

Shuguo Zhang

Keyword(s):

Ideal Convergence ◽

Katětov Order

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On Weakly Tight Families

Canadian Journal of Mathematics ◽

10.4153/cjm-2012-017-8 ◽

2012 ◽

Vol 64 (6) ◽

pp. 1378-1394 ◽

Cited By ~ 5

Author(s):

Dilip Raghavan ◽

Juris Steprāns

Keyword(s):

Disjoint Family ◽

Katětov Order ◽

Almost Disjoint ◽

Almost Disjoint Family ◽

Maximal Almost Disjoint ◽

Almost Disjoint Families

Abstract Using ideas from Shelah's recent proof that a completely separable maximal almost disjoint family exists when 𝔠 < ℵω, we construct a weakly tight family under the hypothesis 𝔰 ≤ 𝔟 < ℵω. The case when 𝔰 < 𝔟 is handled in ZFC and does not require 𝔟 < ℵω, while an additional PCF type hypothesis, which holds when 𝔟 < ℵω is used to treat the case 𝔰 = 𝔟. The notion of a weakly tight family is a natural weakening of the well-studied notion of a Cohen indestructible maximal almost disjoint family. It was introduced by Hrušák and García Ferreira [8], who applied it to the Katétov order on almost disjoint families.

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