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The structure of the clone lattice and Galois connection in multiple-valued logic

Research Project

Project/Area Number 15540112
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field General mathematics (including Probability theory/Statistical mathematics)
Research InstitutionHitotsubashi University

Principal Investigator

MACHIDA Hajime  Hitotsubashi University, Graduate School of Commerce and Management, Professor, 大学院・商学研究科, 教授 (40090534)

Co-Investigator(Kenkyū-buntansha) IWASAKI Shiro  Hitotsubashi University, Graduate School of Economics, Professor, 大学院・経済学研究科, 教授 (00001842)
YAMASAKI Hideki  Hitotsubashi University, Research and Development Center for Higher Education, Professor, 大学教育研究開発センター, 教授 (30108188)
Project Period (FY) 2003 – 2004
Project Status Completed (Fiscal Year 2004)
Budget Amount *help
¥3,500,000 (Direct Cost: ¥3,500,000)
Fiscal Year 2004: ¥1,700,000 (Direct Cost: ¥1,700,000)
Fiscal Year 2003: ¥1,800,000 (Direct Cost: ¥1,800,000)
Keywords(mathematical)clone / lattice of clones / Galois connection / centralizer of a clone / hyperclone
Research Abstract

For a set A, a clone on A is a set of multi-variable functions on A which is closed under composition. The set of all clones on A forms the lattice. We also consider the lattice of all monoids consisting of unary functions, In this research, we considered a naturally defined Galois connection between both lattices. For a monoid M, the centralizer of M is the set of all multi-variable functions which ‘commutes' with all unary functions in M.
1.Some fundamental properties of the Galois connection :
(1)We studied roughly the positions of the centralizers of monoids in the lattice of clones. (2)We showed that for every pair of distinct monoids their centralizers are always distinct.
2.Characterization of the centralizers of the symmetric group and the alternating group :
We established the characterization of the centralizers of both the symmetric group and the alternating group.
3.A sufficient condition for the centralizer of a monoid to be the least clone :
We found a sufficient condition for the centralizer of a monoid to be the least clone which can be used as a very convenient tool.
4.The centralizers of monoids containing the symmetric group :
We determined the centralizers of all monoids which contain the symmetric group. In the course of this research, the above mentioned sufficient condition has been used quite effectively. For most monoids, their centralizers turned out to be the least clone. However, the monoid "M_2" defined over the 4 element set is an exception and its centralizer is not the least clone.
5.Monoids whose centralizer is the least clone :
It is ‘natural' to think that under a Galois connection a small monoid corresponds to a large monoid. However, against this intuition, some small monoids have been discovered whose centralizer is the least clone.

Report

(3 results)
  • 2004 Annual Research Report   Final Research Report Summary
  • 2003 Annual Research Report
  • Research Products

    (22 results)

All 2005 2004 2003 Other

All Journal Article (16 results) Publications (6 results)

  • [Journal Article] Centralizers of monoids containing the symmetric group2005

    • Author(s)
      Machida, H.
    • Journal Title

      Proc. 35th Internat. Symposium on Multiple-Valued Logic 35

      Pages: 227-233

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Centralizers of monoids containing the symmetric group2005

    • Author(s)
      Machida, H., Rosenberg, I.G.
    • Journal Title

      Proc.35th Int.Symp.Multiple-Valued Logic, IEEE

      Pages: 227-233

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Centralizers of monoids containing the symmetric group2005

    • Author(s)
      H.Machida
    • Journal Title

      Proc.35th International Symposium Multiple-Valued Logic 227-233(掲載予定)

    • Related Report
      2004 Annual Research Report
  • [Journal Article] On centralizers of monoids2005

    • Author(s)
      H.Machida
    • Journal Title

      Novi Sad Journal of Mathematics Vol.34, No.2(掲載予定)

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Normal form of hyperoperations and existence of Sheffer hyperoperations2004

    • Author(s)
      Machida, H.
    • Journal Title

      Italian Journal of Pure and Applied Mathematics 16

      Pages: 47-54

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] On centralizers of monoids2004

    • Author(s)
      Machida, H.
    • Journal Title

      Novi Sad Journal of Mathematics 34

      Pages: 153-166

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Monoids whose centralizer is the least clone2004

    • Author(s)
      Machida, H.
    • Journal Title

      Proc. 34th Internat. Symposium on Multiple-Valued Logic 34

      Pages: 102-108

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Normal form of hyperoperations and existence of Sheffer hyperoperations2004

    • Author(s)
      Machida, H.
    • Journal Title

      Italian J.of Pure and Applied Math 16

      Pages: 47-54

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] On centralizers of monoids2004

    • Author(s)
      Machida, H., Rosenberg, I.G.
    • Journal Title

      Novi Sad Journal of Mathematics 34

      Pages: 153-166

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Monoids whose centralizer is the least clone2004

    • Author(s)
      Machida, H., Rosenberg, I.G.
    • Journal Title

      Proc.34th Int.Symp.Multiple-Valued Logic, IEEE

      Pages: 102-108

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Monoids whose centralizer is the least clone2004

    • Author(s)
      H.Machida
    • Journal Title

      Proc.34th International Symposium Multiple-Valued Logic 153-166

      Pages: 102-108

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Normal form of hyperoperations and existence of Sheffer hyperoperations2004

    • Author(s)
      H.Machida
    • Journal Title

      Italian Journal of Pure and Applied Mathematics Vol.16

      Pages: 47-54

    • Related Report
      2004 Annual Research Report
  • [Journal Article] On the centralizers of monoids in clone theory2003

    • Author(s)
      Machida, H.
    • Journal Title

      Proc. 33rd Internat. Symposium on Multiple-Valued Logic 33

      Pages: 303-308

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Some properties of hyperoperations and hyperclones2003

    • Author(s)
      Machida, H.
    • Journal Title

      Words, Languages and Combinatorics III

      Pages: 286-296

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] On the centralizers of monoids in clone theory2003

    • Author(s)
      Machida, H., Rosenberg, I.G.
    • Journal Title

      Proc.33rd Int.Symp.Multiple-Valued Logic, IEEE

      Pages: 303-308

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Some properties of hyperoperations and hyperclones2003

    • Author(s)
      Machida, H.
    • Journal Title

      Words, Languages and Combinatorics III, World Scientific

      Pages: 286-296

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Publications] H.Machida: "Normal form of hyperoperations and existence of Sheffer hyperoperations"Italian Journal of Pure and Applied Mathematics. Vol.19(掲載予定).

    • Related Report
      2003 Annual Research Report
  • [Publications] H.Machida: "Some properties of hyperoperations and hyperclones"Words, Languages and Combinatorics. III. 286-296 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] H.Machida: "Centralizers and monoids in mathematical clone theory"RIMS Kokyuroku. 1325. 146-151 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] H.Machida: "On the centralizers of monoids in clone theory"Proc. 33rd International Symposium on Multiple-Valued Logic. 303-308 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] H.Machida: "Invitation to clone theory--A midsummer night's dream--"Notes on Multiple-Valued Logic in Japan. Vol.26. 2;1-2;4 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] H.Machida: "Monoids whose centralizer is the least clone"Proc. 34th International Symposium on Multiple-Valued Logic. (掲載予定). (2004)

    • Related Report
      2003 Annual Research Report

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Published: 2003-04-01   Modified: 2016-04-21  

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