On Star operation and some ideals on the Baire space

Abstract

We investigate several σ-ideals on the Baire space ω^ω (ℤ^ω), introducing and studying the ideals 𝒢 and 𝒮ℳ𝒵⁺, alongside the classical ideals of meager sets, strong measure zero sets, the eventually different ideal and infinitely equal ideal We establish structural relationships and proper inclusions among these ideals. Also we compute the cardinal invariants of ℳ_-, proving that they are same as invariants of σ-ideal of meager sets. We further analyze the operation ^* on families of sets, establishing dual relationships such as ℰ𝒟^* = ℐℰ, ℐℰ^*=ℰ𝒟, ℳ_-^*=𝒮ℳ𝒵⁺ and ℋ^* = 𝒢, and derive separations between ideals under additional set-theoretic assumptions. Finally, we prove a tree dichotomy theorem for the ideal ℳ_- and we study the associated forcing notion.

Tauberian rates and the Baire property of ideals
Paolo Leonetti
arXiv15 Oct 2026PreprintarXiv licenceFull textAnalysis & PDEsarXiv
On the Average Sizes of Ideals and Antichains in Some Infinite Families of Posets
Miklos Bona, Ricardo Gomez Aiza
arXiv5 Oct 2026PreprintCC BY 4.0Full textCombinatorics & discrete mathematicsarXiv
Analytic multiplier ideals of toric plurisubharmonic functions on a toric complex space
Hoseob Seo
arXiv6 Oct 2026PreprintarXiv licenceFull textAnalysis & PDEsarXiv
Apriori Estimates and Nonexistence Results for Some Quasilinear Problems in a Half-Space
Evgeny Galakhov, Olga Salieva
arXiv29 Aug 2026PreprintCC BY 4.0Full textAnalysis & PDEsarXiv
On some bounds for cardinal invariants of the space of quasicontinuous functions with the topology of pointwise convergence
Mandeep Kumar, Brij Kishore Tyagi
arXiv24 Oct 2026PreprintCC BY 4.0Full textGeometry & topologyarXiv

Cite

BibTeX RIS

BibTeX

@misc{cieslak2026star,
  title = {{On Star operation and some ideals on the Baire space}},
  author = {Aleksander Cieślak and Daria Perkowska and Szymon Żeberski},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2606.01180},
  eprint = {2606.01180},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2606.01180},
}

RIS

TY  - UNPB
TI  - On Star operation and some ideals on the Baire space
AU  - Aleksander Cieślak
AU  - Daria Perkowska
AU  - Szymon Żeberski
PY  - 2026
DA  - 2026/10/30
PB  - arXiv
DO  - 10.48550/arxiv.2606.01180
UR  - https://arxiv.org/abs/2606.01180
AB  - We investigate several σ-ideals on the Baire space ω^ω (ℤ^ω), introducing and studying the ideals 𝒢 and 𝒮ℳ𝒵^+, alongside the classical ideals of meager sets, strong measure zero sets, the eventually different ideal and infinitely equal ideal We establish structural relationships and proper inclusions among these ideals. Also we compute the cardinal invariants of ℳ_-, proving that they are same as invariants of σ-ideal of meager sets. We further analyze the operation ^* on families of sets, establishing dual relationships such as ℰ𝒟^* = ℐℰ, ℐℰ^*=ℰ𝒟, ℳ_-^*=𝒮ℳ𝒵^+ and ℋ^* = 𝒢, and derive separations between ideals under additional set-theoretic assumptions. Finally, we prove a tree dichotomy theorem for the ideal ℳ_- and we study the associated forcing notion.
ER  -

Abstract

Related

Cite