On Farah ideals and beyond
Abstract
An ideal ℐ on ω is called Farah if there are hereditary closed subsets Cn of 𝒫(ω), for n∈ω, such that ℐ={A⊆ω: ∀n∈ω ∃k∈ω A∖ k∈ Cn}. Farah asked in [Trans. Amer. Math. Soc. 356 (2004), no. 6] if every Fσδ ideal is a Farah ideal. We show that this is false by providing both tall and non-tall counterexamples. Moreover, we show that there is an antichain in the sense of Katětov order of length continuum consisting of Fσδ non-Farah ideals. We also investigate a closely related problem concerning sets of ideal limit points. We show that if ℐ is an Gδσδ ideal on ω and x∈ Xω is a sequence in an uncountable Polish space X, then Λx(ℐ)={p∈ X: ∃A∉ℐ xrestriction A converges to p} is Fσ. This answers [Top. App. 312 (2022)] (partially) and [J. Symb. Logic, DOI: 10.1017/jsl.2026.10181]. Earlier it was known to be true only in the case of Farah ideals.
Related
- Closed maximal eventually different families for Σ03 and weakly Farah ideals
- Normality of ideals beyond the standard graded setting: families from numerical semigroup rings
- Asymptotic behaviour of the v-number of homogeneous ideals
- Monomial ideals whose all matching powers are Cohen–Macaulay
- Relative desingularization and principalization of ideals
BibTeX
@misc{kwela2026farah,
title = {{On Farah ideals and beyond}},
author = {Adam Kwela},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.03432},
eprint = {2611.03432},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.03432},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On Farah ideals and beyond
AU - Adam Kwela
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.03432
UR - https://arxiv.org/abs/2611.03432
AB - An ideal ℐ on ω is called Farah if there are hereditary closed subsets C_n of 𝒫(ω), for n∈ω, such that ℐ={A⊆ω: ∀_n∈ω ∃_k∈ω A∖ k∈ C_n}. Farah asked in [Trans. Amer. Math. Soc. 356 (2004), no. 6] if every F_σδ ideal is a Farah ideal. We show that this is false by providing both tall and non-tall counterexamples. Moreover, we show that there is an antichain in the sense of Katětov order of length continuum consisting of F_σδ non-Farah ideals. We also investigate a closely related problem concerning sets of ideal limit points. We show that if ℐ is an G_δσδ ideal on ω and x∈ X^ω is a sequence in an uncountable Polish space X, then Λ_x(ℐ)={p∈ X: ∃_A∉ℐ xrestriction A converges to p} is F_σ. This answers [Top. App. 312 (2022)] (partially) and [J. Symb. Logic, DOI: 10.1017/jsl.2026.10181]. Earlier it was known to be true only in the case of Farah ideals.
ER -