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Structural properties of ideals over P_{kappa}lambda and infinitary combinatorics

Research Project

Project/Area Number 18K03404
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Review Section Basic Section 12030:Basic mathematics-related
Research InstitutionKanagawa University

Principal Investigator

Abe Yoshihiro  神奈川大学, 理学部, 教授 (10159452)

Co-Investigator(Kenkyū-buntansha) 薄葉 季路  早稲田大学, 理工学術院, 教授 (10513632)
南 裕明  愛知学院大学, 教養部, 講師 (70646885)
Project Period (FY) 2018-04-01 – 2023-03-31
Project Status Completed (Fiscal Year 2022)
Budget Amount *help
¥2,730,000 (Direct Cost: ¥2,100,000、Indirect Cost: ¥630,000)
Fiscal Year 2020: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2019: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2018: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
KeywordsPκλ / イデアル / P-point / Q-point / selective ideal / sup-関数 / 定常集合 / Pκλ / 構造的性質 / 非定常イデアル / 正規イデアル / sup-function / 分離基数 / 位相 / extendible 基数 / 強制法 / 飽和度 / non-stationary ideal / local P-point / 基数不変量 / 無限帽子パズル / 集合論的地質学 / 選択公理 / 強コンパクト基数 / weak P-point / 飽和イデアル / Prikry強制法 / Cohen実数 / Pκλ上のイデアル / non-stationary イデアル / 弱い分割の性質
Outline of Final Research Achievements

We study weak notions of selectivity and show that a nowher P-point exists above the bounded ideal in the Rudin-Keisler ordering. It is also proved that an ideal is not a local P-point if and only if it includes a certain type of ideals.
About a stationary subet of P_{\kappa}\lambda on which the sup-function is one-to-one, we define a combinatorial principle which implies the existence of such a set and construct the forcing models where it exists and it does not exist. Relations between the saturation of the restriction of the non-stationary ideal and the existence of such a set is turned out.

Academic Significance and Societal Importance of the Research Achievements

正則基数κ上のイデアルの構造的性質に関しては、既に1980年代に詳しく調べられていたが、Pκλ上のイデアルについては、過去2回の助成研究以前は殆ど結果らしいものはなかった。前々回で基本概念を定義し、強いselectivityに関しては前回の研究でかなりの事を明らかにできたが、弱いselectivityに関しては、手が付けられていなかった。今回、突破口となる結果がいくつか得られ、今後の進展に寄与すると思われる。
前回、多くの結果はsup-関数が1対1という仮定の下で得られていたが、そのような定常集合の存在について、無矛盾性の強さも含めて多数の事実を明らかにできたのは、高く評価されると自負している。

Report

(6 results)
  • 2022 Annual Research Report   Final Research Report ( PDF )
  • 2021 Research-status Report
  • 2020 Research-status Report
  • 2019 Research-status Report
  • 2018 Research-status Report
  • Research Products

    (41 results)

All 2023 2022 2021 2020 2019 2018

All Journal Article (16 results) (of which Peer Reviewed: 13 results,  Open Access: 3 results) Presentation (25 results) (of which Int'l Joint Research: 9 results,  Invited: 6 results)

  • [Journal Article] The coloring number and the list-chromatic number of uncountable graphs2023

    • Author(s)
      Toshimichi Usuba
    • Journal Title

      Israel Journal of Mathematics

      Volume: -

    • Related Report
      2022 Annual Research Report 2021 Research-status Report
    • Peer Reviewed
  • [Journal Article] C*-embedding and P-embedding in subspaces of products of ordinals2022

    • Author(s)
      Nobuyuki Kemoto、Toshimichi Usuba
    • Journal Title

      Topology and its Applications

      Volume: 318 Pages: 108194-108194

    • DOI

      10.1016/j.topol.2022.108194

    • Related Report
      2022 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Lowenheim-Skolem-Tarski の性質2022

    • Author(s)
      薄葉季路
    • Journal Title

      数理解析研究所考究録

      Volume: 2233 Pages: 82-104

    • Related Report
      2022 Annual Research Report
  • [Journal Article] Equalities for the extent of infinite products and Σ-products2022

    • Author(s)
      Yasushi Hirata,Toshimichi Usuba, Yukinobu Yajima
    • Journal Title

      Topology and its Applications

      Volume: 307

    • Related Report
      2021 Research-status Report
    • Peer Reviewed
  • [Journal Article] 無限積空間とΣ-積空間における基数関数extentの等式について2022

    • Author(s)
      平田康史、薄葉季路、矢島幸信
    • Journal Title

      数理解析研究所講究録

      Volume: 2209 Pages: 64-74

    • Related Report
      2021 Research-status Report
    • Open Access
  • [Journal Article] Geology of symmetric grounds2021

    • Author(s)
      Toshimichi Usuba
    • Journal Title

      Proceedings of Asian Logic Conference 2019

      Volume: to appear

    • Related Report
      2020 Research-status Report
    • Peer Reviewed
  • [Journal Article] A note on δ-strongly compact cardinals2020

    • Author(s)
      Usuba Toshimichi
    • Journal Title

      Topology and its Applications

      Volume: - Pages: 107538-107538

    • DOI

      10.1016/j.topol.2020.107538

    • Related Report
      2020 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] A note on Lowenheim-Skolem cardinals2020

    • Author(s)
      Toshimichi Usuba
    • Journal Title

      数理解析研究所講究録

      Volume: 2164 Pages: 149-160

    • NAID

      120006956550

    • Related Report
      2020 Research-status Report
    • Open Access
  • [Journal Article] On Countable Stationary Towers.2020

    • Author(s)
      Yo Matsubara, Toshimichi Usuba,
    • Journal Title

      Proceedings of Symposium on Advances in Mathematical Logic 2018.

      Volume: -

    • Related Report
      2019 Research-status Report
    • Peer Reviewed
  • [Journal Article] Choiceless L\"owenheim-Skolem property and uniform definability of grounds.2020

    • Author(s)
      Toshimichi Usuba,
    • Journal Title

      Proceedings of Symposium on Advances in Mathematical Logic 2018.

      Volume: -

    • Related Report
      2019 Research-status Report
    • Peer Reviewed
  • [Journal Article] A note on the tightness of G-modifications2019

    • Author(s)
      Usuba Toshimichi
    • Journal Title

      Topology and its Applications

      Volume: 265 Pages: 106820-106820

    • DOI

      10.1016/j.topol.2019.106820

    • Related Report
      2019 Research-status Report
    • Peer Reviewed
  • [Journal Article] Products of Lindeloef spaces with points $G_delta$2019

    • Author(s)
      Usuba Toshimichi
    • Journal Title

      Topology and its Applications

      Volume: 252 Pages: 90-96

    • DOI

      10.1016/j.topol.2018.11.019

    • Related Report
      2018 Research-status Report
    • Peer Reviewed
  • [Journal Article] On the existence of skinny stationary subsets2019

    • Author(s)
      Matsubara Yo、Sakai Hiroshi、Usuba Toshimichi
    • Journal Title

      Annals of Pure and Applied Logic

      Volume: 170 Issue: 5 Pages: 539-557

    • DOI

      10.1016/j.apal.2018.12.003

    • NAID

      120006604682

    • Related Report
      2018 Research-status Report
    • Peer Reviewed
  • [Journal Article] $G_\delta $-topology and compact cardinals2019

    • Author(s)
      Usuba Toshimichi
    • Journal Title

      Fundamenta Mathematicae

      Volume: 246 Issue: 1 Pages: 71-87

    • DOI

      10.4064/fm487-7-2018

    • Related Report
      2018 Research-status Report
    • Peer Reviewed
  • [Journal Article] New combinatorial principle on singular cardinals and normal ideals2018

    • Author(s)
      Usuba Toshimichi
    • Journal Title

      Mathematical Logic Quarterly

      Volume: 64 Issue: 4-5 Pages: 395-408

    • DOI

      10.1002/malq.201700024

    • Related Report
      2018 Research-status Report
    • Peer Reviewed
  • [Journal Article] Extendible cardinals and the mantle2018

    • Author(s)
      Usuba Toshimichi
    • Journal Title

      Archive for Mathematical Logic

      Volume: 58 Issue: 1-2 Pages: 71-75

    • DOI

      10.1007/s00153-018-0625-4

    • Related Report
      2018 Research-status Report
    • Peer Reviewed
  • [Presentation] 可算集合の可算和について(ただし選択公理は仮定しない)2023

    • Author(s)
      薄葉季路
    • Organizer
      第8回山陰基礎論と数学およびその周辺の研究集会
    • Related Report
      2022 Annual Research Report
  • [Presentation] Monotonicity of ultrafilter numbers2023

    • Author(s)
      薄葉季路
    • Organizer
      日本数学会2023年度年会
    • Related Report
      2022 Annual Research Report
  • [Presentation] Lowenheim-Skolem-Tarski property of rings of continuous functions2022

    • Author(s)
      薄葉季路
    • Organizer
      Symposium on Advances in Mathematical Logic
    • Related Report
      2022 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] The continuum function on the countable unions of countable sets2022

    • Author(s)
      宮内凛、薄葉季路
    • Organizer
      2022年度日本数学会秋季総合分科会
    • Related Report
      2022 Annual Research Report
  • [Presentation] Generically extendible cardinals2021

    • Author(s)
      薄葉季路
    • Organizer
      日本数学会2021 年度秋季総合分科会
    • Related Report
      2021 Research-status Report
  • [Presentation] Equalities for the extent of infinite products and Σ-products2021

    • Author(s)
      矢島幸信, 平田康史, 薄葉季路
    • Organizer
      日本数学会2021 年度秋季総合分科会
    • Related Report
      2021 Research-status Report
  • [Presentation] Generic setwise large cardinals2021

    • Author(s)
      Toshimichi Usuba
    • Organizer
      Kobe Set Theory Workshop 2021
    • Related Report
      2020 Research-status Report
    • Int'l Joint Research
  • [Presentation] Variants of strong Chang's conjecture2021

    • Author(s)
      薄葉季路
    • Organizer
      日本数学会2021年度年会
    • Related Report
      2020 Research-status Report
  • [Presentation] On Reinhardt cardinals2020

    • Author(s)
      薄葉季路
    • Organizer
      日本数学会2020年度秋季総合分科会
    • Related Report
      2020 Research-status Report
  • [Presentation] On generically extendible cardinals2020

    • Author(s)
      薄葉季路
    • Organizer
      日本数学会2020年度年会
    • Related Report
      2019 Research-status Report
  • [Presentation] A condition for an ideal to be a P-poit2019

    • Author(s)
      阿部吉弘
    • Organizer
      日本数学会秋季総合分科会
    • Related Report
      2019 Research-status Report
  • [Presentation] Choiceless set-theoretic geology2019

    • Author(s)
      Toshimichi Usuba
    • Organizer
      Higher recursion theory and set theory
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Choiceless set-theoretic geology2019

    • Author(s)
      Toshimichi Usuba
    • Organizer
      Asian Logic Conference 2019
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Standard and choiceless set-theoretic geology2019

    • Author(s)
      薄葉季路
    • Organizer
      Summer School 2019 (東北大学)
    • Related Report
      2019 Research-status Report
  • [Presentation] Large cardinals as natural upper bounds on cardinal functions2019

    • Author(s)
      Toshimichi Usuba
    • Organizer
      3rd Pan-Pacific International Conference on Topology and Applications
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Geology of symmetric grounds2019

    • Author(s)
      Toshimichi Usuba
    • Organizer
      Set Theory and Infinity
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research
  • [Presentation] 多元宇宙の理論2019

    • Author(s)
      薄葉季路
    • Organizer
      山陰基礎論・解析学セミナー
    • Related Report
      2018 Research-status Report
  • [Presentation] GCH at strongly compact cardinals2019

    • Author(s)
      薄葉季路
    • Organizer
      日本数学会2019年度年会
    • Related Report
      2018 Research-status Report
  • [Presentation] The generic multiverse and large cardinals2018

    • Author(s)
      Toshimichi Usuba
    • Organizer
      Symposium on Advances in Mathematical Logic 2018
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research
  • [Presentation] ω1-strongly compact cardinals and cardinal functions in topology2018

    • Author(s)
      Toshimichi Usuba
    • Organizer
      Reflections on Set Theoreric Reflection
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research
  • [Presentation] Extendible cardinals and the mantle2018

    • Author(s)
      薄葉季路
    • Organizer
      日本数学会秋季総合分科会
    • Related Report
      2018 Research-status Report
  • [Presentation] Syntactical and semantical approachs to generic multiverse2018

    • Author(s)
      Toshimichi Usuba
    • Organizer
      Sendai Logic School 2018
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Gδ-topology and compact cardinals2018

    • Author(s)
      薄葉季路
    • Organizer
      一般位相幾何学の進展と諸問題
    • Related Report
      2018 Research-status Report
  • [Presentation] 数学基礎論と位相空間論のコンパクト2018

    • Author(s)
      薄葉季路
    • Organizer
      筑波大学数学域談話会
    • Related Report
      2018 Research-status Report
    • Invited
  • [Presentation] Axiomatization of Generic Multiverse2018

    • Author(s)
      薄葉季路
    • Organizer
      証明論研究集会
    • Related Report
      2018 Research-status Report

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Published: 2018-04-23   Modified: 2024-01-30  

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