Filters, ideal independence and ideal Mrówka spaces

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F67985840%3A_____%2F25%3A00637584" target="_blank" >RIV/67985840:_____/25:00637584 - isvavai.cz</a>

Výsledek na webu

<a href="https://doi.org/10.1090/tran/9450" target="_blank" >https://doi.org/10.1090/tran/9450</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1090/tran/9450" target="_blank" >10.1090/tran/9450</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Filters, ideal independence and ideal Mrówka spaces

Popis výsledku v původním jazyce

A family A ⊆ [ω]ω such that for all finite {Xi}i∈n ⊆ A and A ∈ A{Xi}i∈n, the set AUi∈n Xi is infinite, is said to be ideal independent. We prove that an ideal independent family A is maximal if and only if A is J-completely separable and maximal J-almost disjoint for a particular ideal J on ω. We show that u ≤ smm, where smm is the minimal cardinality of maximal ideal independent family. This, in particular, establishes the independence of smm and i. Given an arbitrary set C of uncountable cardinals, we show how to simultaneously adjoin via forcing maximal ideal independent families of cardinality λ for each λ ∈ C, thus establishing the consistency of C ⊆ spec(smm). Assuming CH, we construct a maximal ideal independent family, which remains maximal after forcing with any proper, ωω-bounding, p-point preserving forcing notion and evaluate smm in several well-studied forcing extensions. We also study natural filters associated with ideal independence and introduce an analog of Mrówka spaces for ideal independent families.

Název v anglickém jazyce

Filters, ideal independence and ideal Mrówka spaces

Popis výsledku anglicky

A family A ⊆ [ω]ω such that for all finite {Xi}i∈n ⊆ A and A ∈ A{Xi}i∈n, the set AUi∈n Xi is infinite, is said to be ideal independent. We prove that an ideal independent family A is maximal if and only if A is J-completely separable and maximal J-almost disjoint for a particular ideal J on ω. We show that u ≤ smm, where smm is the minimal cardinality of maximal ideal independent family. This, in particular, establishes the independence of smm and i. Given an arbitrary set C of uncountable cardinals, we show how to simultaneously adjoin via forcing maximal ideal independent families of cardinality λ for each λ ∈ C, thus establishing the consistency of C ⊆ spec(smm). Assuming CH, we construct a maximal ideal independent family, which remains maximal after forcing with any proper, ωω-bounding, p-point preserving forcing notion and evaluate smm in several well-studied forcing extensions. We also study natural filters associated with ideal independence and introduce an analog of Mrówka spaces for ideal independent families.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

—

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2025

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

American Mathematical Society. Transactions

ISSN

0002-9947

e-ISSN

1088-6850

Svazek periodika

378

Číslo periodika v rámci svazku

8

Stát vydavatele periodika

US - Spojené státy americké

Počet stran výsledku

17

Strana od-do

5423-5439

Kód UT WoS článku

001544345000007

EID výsledku v databázi Scopus

2-s2.0-105010104737