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TUKEY ORDER AMONG $F_{\sigma }$ IDEALS

Part of: Ordered sets Set theory

Published online by Cambridge University Press:  06 May 2021

JIALIANG HE ,
MICHAEL HRUŠÁK ,
DIEGO ROJAS-REBOLLEDO  and
SŁAWOMIR SOLECKI
JIALIANG HE
Affiliation:
COLLEGE OF MATHEMATICSSICHUAN UNIVERSITYCHENGDU, SICHUAN610064, CHINAE-mail: jialianghe@scu.edu.cn
MICHAEL HRUŠÁK
Affiliation:
CENTRO DE CIENCIAS MATEMÁTICAS UNIVERSIDAD NACIONAL AUTÓNOMA DE MÉXICOMORELIA, MEXICOE-mail: michael@matmor.unam.mx
DIEGO ROJAS-REBOLLEDO
Affiliation:
DEPARTMENT OF MATHEMATICS AND COMPUTING SCIENCES SAINT MARY’S UNIVERSITYHALIFAX, NS, CANADAE-mail: Diego.Rojas@smu.ca
SŁAWOMIR SOLECKI
Affiliation:
DEPARTMENT OF MATHEMATICS CORNELL UNIVERSITYITHACA, NY14853, USAE-mail: ssolecki@cornell.edu
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Abstract

We investigate the Tukey order in the class of $F_{\sigma }$ ideals of subsets of $\omega $ . We show that no nontrivial $F_{\sigma }$ ideal is Tukey below a $G_{\delta }$ ideal of compact sets. We introduce the notions of flat ideals and gradually flat ideals. We prove a dichotomy theorem for flat ideals isolating gradual flatness as the side of the dichotomy that is structurally good. We give diverse characterizations of gradual flatness among flat ideals using Tukey reductions and games. For example, we show that gradually flat ideals are precisely those flat ideals that are Tukey below the ideal of density zero sets.

Keywords

Tukey orderFσidealflat idealgradually flat ideal

MSC classification

Primary: 03E15: Descriptive set theory
Secondary: 03E05: Other combinatorial set theory 06A07: Combinatorics of partially ordered sets
Type
Article
Information
The Journal of Symbolic Logic , Volume 86 , Issue 2 , June 2021 , pp. 855 - 870
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Copyright
© The Association for Symbolic Logic 2021

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