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Classification of minimal clones over a finite field in multiple-valued logic

Research Project

Project/Area Number 20540111
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, 大学院・アーツ・サイエンス研究科, 研究員 (40090534)

Co-Investigator(Kenkyū-buntansha) YAMASAKI Hideki  一橋大学, 大学院・商学研究科, 教授 (30108188)
Co-Investigator(Renkei-kenkyūsha) POGOSYAN Grant  国際基督教大学, 教養学部, 教授 (90234640)
Project Period (FY) 2008 – 2010
Project Status Completed (Fiscal Year 2010)
Budget Amount *help
¥4,420,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥1,020,000)
Fiscal Year 2010: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Fiscal Year 2009: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000)
Fiscal Year 2008: ¥2,080,000 (Direct Cost: ¥1,600,000、Indirect Cost: ¥480,000)
Keywords離散数学 / 普遍代数 / 多値論理
Research Abstract

For a fixed set A, a set of multi-variable functions defined over A is called a clone if it contains all projections and is closed with respect to (functional) composition. The set of all clones on A has the structure of a lattice. An atom of the lattice of clones is called a minimal clone. In this research, we introduced the structure of a finite field into the base set A and considered generating functions of minimal clones as polynomials over the finite field A. Under this new point of view, we aimed at classifying minimal clones.
For minimal clones which are generated by binary idempotent functions or ternary majority functions, we have succeeded to find the polynomial expressions of their minimal functions for arbitrary A with the cardinality of prime power.

Report

(4 results)
  • 2010 Annual Research Report   Final Research Report ( PDF )
  • 2009 Annual Research Report
  • 2008 Annual Research Report
  • Research Products

    (56 results)

All 2011 2010 2009 2008 Other

All Journal Article (28 results) (of which Peer Reviewed: 23 results) Presentation (28 results)

  • [Journal Article] Completeness of order algebras2011

    • Author(s)
      I. Chajda, R. Halas, H. Machida
    • Journal Title

      Journal of Multiple-Valued Logic and Soft Computing Vol. 17

      Pages: 93-98

    • Related Report
      2010 Final Research Report
    • Peer Reviewed
  • [Journal Article] Completeness of order algebras2011

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

      Journal of Multiple-Valued Logic and Soft Computing

      Volume: 17 Pages: 93-98

    • Related Report
      2010 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Commutation and centralizers in clone theory2010

    • Author(s)
      H. Machida
    • Journal Title

      京都大学数理解析研究所 講究録 Vol. 1712

      Pages: 101-110

    • Related Report
      2010 Final Research Report
  • [Journal Article] Endoprimal monoids and witness lemma in clone theory2010

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

      Proceedings 40th International Symposium on Multiple--Valued Logic, IEEE 40

      Pages: 195-200

    • Related Report
      2010 Final Research Report
    • Peer Reviewed
  • [Journal Article] Endoprimal monoids and witness lemma in clone theory2010

    • Author(s)
      H.Machida
    • Journal Title

      Proceedings 40th International Symposium on Multiple-Valued Logic

      Volume: 40 Pages: 195-200

    • Related Report
      2010 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Galois connections for hyperclones2010

    • Author(s)
      H.Machida
    • Journal Title

      Proceedings 40th International Symposium on Multiple-Valued Logic

      Volume: 40 Pages: 201-204

    • Related Report
      2010 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Commutation and centralizers in clone theory2010

    • Author(s)
      H.Machida
    • Journal Title

      京都大学数理解析研究所 講究録

      Volume: 1712 Pages: 101-110

    • Related Report
      2010 Annual Research Report
  • [Journal Article] Completeness of order algebras2010

    • Author(s)
      I.Chajda
    • Journal Title

      Journal of Multiple-Valued Logic and Soft Computing (掲載決定)

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Three classes of maximal hyperclones2010

    • Author(s)
      H.Machida
    • Journal Title

      Journal of Multiple-Valued Logic and Soft Computing (掲載決定)

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Galois connections for hyperclones2010

    • Author(s)
      H.Machida
    • Journal Title

      Proceedings 40th International Symposium on Multiple-Valued Logic 40(掲載決定)

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Endoprimal monoids and witness lemma in clone theory2010

    • Author(s)
      H.Machida
    • Journal Title

      Proceedings 40th International Symposium on Multiple-Valued Logic 40(掲載決定)

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Maximal hyperclones on E_2 as hypercores2009

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

      Journal of Multiple-Valued Logic and Soft Computing Vol. 15

      Pages: 315-328

    • Related Report
      2010 Final Research Report
    • Peer Reviewed
  • [Journal Article] Proceedings 39th International Symposium on Multiple-Valued Logic, IEEE2009

    • Author(s)
      H. Machida, J. Pantovic, Hyperclones determined by total parts of hyperrelations
    • Journal Title

      39 161-166

      Pages: 1-1

    • Related Report
      2010 Final Research Report
  • [Journal Article] On endoprimal monoids in clone theory2009

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

      Proceedings 39th International Symposium on Multiple-Valued Logic, IEEE 39

      Pages: 167-172

    • Related Report
      2010 Final Research Report
    • Peer Reviewed
  • [Journal Article] Maximal hyperclones on E_2 as hypercores2009

    • Author(s)
      H.Machida
    • Journal Title

      Journal of Multiple-Valued Logic and Soft Computing Vol. 15

      Pages: 315-328

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Hyperclones determined by total-parts of hyper-relations2009

    • Author(s)
      H.Machida
    • Journal Title

      Proceedings 39th International Symposium on Multiple-Valued Logic 39

      Pages: 161-166

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] On endoprimal monoids in clone theory2009

    • Author(s)
      H.Machida
    • Journal Title

      Proceedings 39th International Symposium on Multiple-Valued Logic 39

      Pages: 167-172

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Majority and other polynomials in minimal clones2008

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

      Proceedings 38th International Symposium on Multiple-Valued Logic, IEEE 38

      Pages: 38-43

    • Related Report
      2010 Final Research Report
    • Peer Reviewed
  • [Journal Article] On the isomorphism problem of concept algebras2008

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

      Proceedings of the 6th International Conference on Concept Lattices and Their Applications, Palacky University(Czech Republic) 6

      Pages: 217-228

    • Related Report
      2010 Final Research Report
    • Peer Reviewed
  • [Journal Article] On minimal clones and generating polynomials2008

    • Author(s)
      H. Machida
    • Journal Title

      京都大学数理解析研究所 講究録 Vol. 1599

      Pages: 35-41

    • Related Report
      2010 Final Research Report
  • [Journal Article] On maximal hyperclones on {0, 1} -- A new approach --2008

    • Author(s)
      H. Machida
    • Journal Title

      Proceedings 38th International Symposium on Multiple-Valued Logic 38

      Pages: 32-37

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Majority and Other Polynomials in Minimal Clones2008

    • Author(s)
      H. Machida
    • Journal Title

      Proceedings 38th International Symposium on Multiple-Valued Logic 38

      Pages: 38-43

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Journal Article] On the isomorphism problem of concept algebras2008

    • Author(s)
      L. Kwuida
    • Journal Title

      Proceedings of the 6th International Conference on Concept Lattices and Their Applications (CLA 2008) 6

      Pages: 217-228

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Journal Article] On Minimal Clones and Generating Polynomials2008

    • Author(s)
      H. Machida
    • Journal Title

      京都大学数理解析研究所 講究録 1599

      Pages: 35-41

    • Related Report
      2008 Annual Research Report
  • [Journal Article] Some centralizing monoids in clone theory

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

      Journal of Multiple-Valued Logic and Soft Computing (掲載決定)

    • Related Report
      2010 Final Research Report
    • Peer Reviewed
  • [Journal Article] Three classes of maximal hyperclones

    • Author(s)
      H.Machida
    • Journal Title

      Journal of Multiple-Valued Logic and Soft Computing

      Volume: (掲載決定)

    • Related Report
      2010 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Some centralizing monoids in clone theory

    • Author(s)
      H.Machida
    • Journal Title

      Journal of Multiple-Valued Logic and Soft Computing

      Volume: (掲載決定)

    • Related Report
      2010 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Maximal hyperclones on E_2 as hypercores(掲載決定)

    • Author(s)
      H. Machida
    • Journal Title

      Journal of Multiple-Valued Logic and Soft Computing (掲載決定)

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Presentation] Centralizing monoids and minimal clones2011

    • Author(s)
      H. Machida
    • Organizer
      京都大学数理解析研究所 RIMS研究集会"Algorithm and Computation Theory in Algebras and Languages"
    • Place of Presentation
      京都大学
    • Year and Date
      2011-02-23
    • Related Report
      2010 Final Research Report
  • [Presentation] Centralizing monoids and minimal clones2011

    • Author(s)
      H.Machida
    • Organizer
      RIMS 研究集会 "Algorithm and Computation Theory in Algebras and Languages"
    • Place of Presentation
      京都大学数理解析研究所
    • Year and Date
      2011-02-23
    • Related Report
      2010 Annual Research Report
  • [Presentation] Centralizing monoids, witnesses and minimal clones2010

    • Author(s)
      H. Machida
    • Organizer
      Summer School on General Algebra and Ordered Sets
    • Place of Presentation
      Malenovice (チェコ共和国)(招待講演)
    • Year and Date
      2010-09-05
    • Related Report
      2010 Final Research Report
  • [Presentation] Centralizing monoids, witnesses and minimal clones2010

    • Author(s)
      H.Machida
    • Organizer
      Summer School on General Algebra and Ordered Sets
    • Place of Presentation
      Malenovice(チェコ共和国)(招待講演)
    • Year and Date
      2010-09-05
    • Related Report
      2010 Annual Research Report
  • [Presentation] Galois connections for hyperclones2010

    • Author(s)
      H. Machida, J. Pantovic, I. G. Rosenberg
    • Organizer
      The Fortieth International Symposium on Multiple-Valued Logic
    • Place of Presentation
      Balcelona (スペイン)
    • Year and Date
      2010-05-27
    • Related Report
      2010 Final Research Report
  • [Presentation] Endoprimal monoids and witness lemma in clone theory2010

    • Author(s)
      H. Machida, I. G. Rosenberg
    • Organizer
      The Fortieth International Symposium on Multiple-Valued Logic
    • Place of Presentation
      Balcelona (スペイン)
    • Year and Date
      2010-05-27
    • Related Report
      2010 Final Research Report
  • [Presentation] Endoprimal monoids and witness lemma in clone theory2010

    • Author(s)
      H.Machida
    • Organizer
      The Fortieth International Symposium on Multiple-Valued Logic
    • Place of Presentation
      Barcelona (スペイン)
    • Year and Date
      2010-05-27
    • Related Report
      2010 Annual Research Report
  • [Presentation] Galois connections for hyperclones2010

    • Author(s)
      H.Machida
    • Organizer
      The Fortieth International Symposium on Multiple-Valued Logic
    • Place of Presentation
      Barcelona (スペイン)
    • Year and Date
      2010-05-27
    • Related Report
      2010 Annual Research Report
  • [Presentation] Centralizers of monoids and monoids of centralizers2010

    • Author(s)
      H.Machida
    • Organizer
      Wiener Algebra Seminar
    • Place of Presentation
      ウィーン工科大学
    • Year and Date
      2010-03-19
    • Related Report
      2009 Annual Research Report
  • [Presentation] Commutation and centralizers in clone theory2010

    • Author(s)
      H. Machida
    • Organizer
      京都大学数理解析研究所 RIMS 研究集会"Algorithm and Computation Theory inAlgebras and Languages"
    • Place of Presentation
      京都大学(招待講演)
    • Year and Date
      2010-02-18
    • Related Report
      2010 Final Research Report
  • [Presentation] Commutation and centralizers in clone theory(招待講演)2010

    • Author(s)
      H.Machida
    • Organizer
      RIMS研究集会"Algorithm and Computation Theory in Algebras and Languages"
    • Place of Presentation
      京都大学
    • Year and Date
      2010-02-18
    • Related Report
      2009 Annual Research Report
  • [Presentation] Essentially minimal clones : Retrospective and prospective2009

    • Author(s)
      H. Machida
    • Organizer
      京都大学数理解析研究所 RIMS 研究集会"Algebra and Logic Related to Computer Science"
    • Place of Presentation
      京都大学
    • Year and Date
      2009-10-23
    • Related Report
      2010 Final Research Report
  • [Presentation] Essentially minimal clones : Retrospective and prospective2009

    • Author(s)
      H.Machida
    • Organizer
      RIMS研究集会"Algebra and Logic Related to Computer Science"
    • Place of Presentation
      京都大学
    • Year and Date
      2009-10-23
    • Related Report
      2009 Annual Research Report
  • [Presentation] Endoprimal monoids, commutation and witness2009

    • Author(s)
      H. Machida
    • Organizer
      Summer School on General Algebra and Ordered Sets
    • Place of Presentation
      Stara Lesna (スロバキア)
    • Year and Date
      2009-09-06
    • Related Report
      2010 Final Research Report
  • [Presentation] Endoprimal monoids, commutation and witness2009

    • Author(s)
      H.Machida
    • Organizer
      Summer School on General Algebra and Ordered Sets
    • Place of Presentation
      スロバキア Stars Lesna 市
    • Year and Date
      2009-09-06
    • Related Report
      2009 Annual Research Report
  • [Presentation] On endoprimal monoids in clone theory2009

    • Author(s)
      H. Machida, I. G. Rosenberg
    • Organizer
      The Thirty Ninth International Symposium on Multiple-Valued Logic
    • Place of Presentation
      沖縄県那覇市
    • Year and Date
      2009-05-22
    • Related Report
      2010 Final Research Report
  • [Presentation] On endoprimal monoids in clone theory2009

    • Author(s)
      H.Machida
    • Organizer
      39th International Symposium on Multiple-Valued Logic
    • Place of Presentation
      沖縄県 那覇市
    • Year and Date
      2009-05-22
    • Related Report
      2009 Annual Research Report
  • [Presentation] Some endoprimal monoids over a three-element set2009

    • Author(s)
      H. Machida
    • Organizer
      77th Workshop on General Algebra (AAA77)
    • Place of Presentation
      Potsdam (ドイツ)
    • Year and Date
      2009-03-20
    • Related Report
      2010 Final Research Report
  • [Presentation] Some endoprimal monoids over a three-element set2009

    • Author(s)
      H. Machida
    • Organizer
      77th Workshop on General Algebra(AAA77)
    • Place of Presentation
      ドイツPotsdam市Potsdam大学
    • Year and Date
      2009-03-20
    • Related Report
      2008 Annual Research Report
  • [Presentation] Some endoprimal monoids of 3-valued unary functions2009

    • Author(s)
      H. Machida
    • Organizer
      第22回多値論理とその応用研究会
    • Place of Presentation
      群馬大学 工学部
    • Year and Date
      2009-01-10
    • Related Report
      2008 Annual Research Report
  • [Presentation] On minimal clones and polynomials over GF(k)2008

    • Author(s)
      H. Machida
    • Organizer
      Summer School on General Algebra and Ordered Sets
    • Place of Presentation
      Trest (チェコ共和国)
    • Year and Date
      2008-09-01
    • Related Report
      2010 Final Research Report
  • [Presentation] On minimal clones and polynomials over GF (k)2008

    • Author(s)
      H. Machida
    • Organizer
      Summer School on General Algebra and Ordered Sets (SSAOS-2008)
    • Place of Presentation
      チェコ共和国Trest市Castle Hotel
    • Year and Date
      2008-09-01
    • Related Report
      2008 Annual Research Report
  • [Presentation] Minimal clones on finite fields2008

    • Author(s)
      H. Machida
    • Organizer
      京都大学数理解析研究所 RIMS研究集会"Clone Theory and Discrete Mathematics"
    • Place of Presentation
      京都大学
    • Year and Date
      2008-07-30
    • Related Report
      2010 Final Research Report
  • [Presentation] Minimal clones on finite fields2008

    • Author(s)
      H. Machida
    • Organizer
      京都大学数理解析研究所RIMS研究集会
    • Place of Presentation
      京都大学数理解析研究所
    • Year and Date
      2008-07-30
    • Related Report
      2008 Annual Research Report
  • [Presentation] Majority and other polynomials in minimal clones2008

    • Author(s)
      H. Machida, T. Waldhauser
    • Organizer
      The Thirty Eighth International Symposium on Multiple-Valued Logic
    • Place of Presentation
      Dallas (アメリカ合衆国)
    • Year and Date
      2008-05-22
    • Related Report
      2010 Final Research Report
  • [Presentation] On maximal hyperclones on {0,1} -- A new approach--2008

    • Author(s)
      H. Machida, J. Pantovic
    • Organizer
      The Thirty Eighth International Symposium on Multiple-Valued Logic
    • Place of Presentation
      Dallas (アメリカ合衆国)("Outstanding Paper Award" 受賞)
    • Year and Date
      2008-05-22
    • Related Report
      2010 Final Research Report
  • [Presentation] On maximal hyperclones on {0, 1} -- A new approach --2008

    • Author(s)
      H. Machida
    • Organizer
      Thirty Eighth International Symposium on Multiple-Valued Logic (ISMVL-2008)
    • Place of Presentation
      米国Dallas市Southern Methodist大学
    • Year and Date
      2008-05-22
    • Related Report
      2008 Annual Research Report
  • [Presentation] Majority and Other Polynonmials in Minimal Clones2008

    • Author(s)
      H. Machida
    • Organizer
      Thirty Eighth International Symposium on Multiple-Valued Logic (ISMVL-2008)
    • Place of Presentation
      米国Dallas市Southern Methodist大学
    • Year and Date
      2008-05-22
    • Related Report
      2008 Annual Research Report

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Published: 2008-04-01   Modified: 2016-04-21  

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