Project/Area Number  10640109 
Research Category 
GrantinAid for Scientific Research (C).

Section  一般 
Research Field 
General mathematics (including Probability theory/Statistical mathematics)

Research Institution  Hitotsubashi University 
Principal Investigator 
MACHIDA Hajime Hitotsubashi Univ. Dept. of Commerce, Professor, 商学部, 教授 (40090534)

CoInvestigator(Kenkyūbuntansha) 
ISHIMURA Naoyuki Grad. S., Economics, Assoc. Prof., 大学院・経済学研究科, 助教授 (80212934)
YAMAZAKI Masao Grad. S., Economics, Assoc. Prof., 大学院・経済学研究科, 助教授 (20174659)
FUJITA Takahiko Dept. of Commerce, Professor, 商学部, 教授 (50144316)
YAMADA Hiromochi Grad. S., Economics, Professor, 大学院・経済学研究科, 教授 (50134888)
山崎 秀記 一橋大学, 商学部, 教授 (30108188)
高岡 浩一郎 一橋大学, 商学部, 講師 (50272662)
IWASAKI Shiro Grad. S., Economics, Professor, 大学院・経済学研究科, 教授 (00001842)

Project Fiscal Year 
1998 – 1999

Project Status 
Completed(Fiscal Year 1999)

Budget Amount *help 
¥2,200,000 (Direct Cost : ¥2,200,000)
Fiscal Year 1999 : ¥1,000,000 (Direct Cost : ¥1,000,000)
Fiscal Year 1998 : ¥1,200,000 (Direct Cost : ¥1,200,000)

Keywords  multiplevalued logical function / (mathematical) clone / lattice of clones / minimal clone / hyperclone / 多値論理関数 / (数学的)クローン理論 / クローン束 / 極小クローン / 超クローン / クローン理論 
Research Abstract 
A clone is a set of kvalued logical functions which is closed under composition and contains all the projections. The set of all clones consisting of kvalued logical functions is denoted by LィイD2kィエD2. Whereas the structure of LィイD22ィエD2 is completely determined, our knowledge about the structure of LィイD2kィエD2 for k > 2 at present is very little. The main objective of this research is to clarify the structure of LィイD2kィエD2 and we have obtained the following results. 1. The structure of LィイD23ィエD2 as a metric space We have introduced a metric into the lattice LィイD2kィエD2 of clones and showed that LィイD2kィエD2 is a compact metric space. Moreover, we constructed continuous mappings, based on the meet operation, from LィイD23ィエD2 onto LィイD22ィエD2 and studied the images of maximal clones in LィイD23ィエD2 and those of some clones being accumulation points under such mappings. 2. Minimal clones in LィイD2kィエD2 and related topics Since the classification of minimal clones is far from complete, the study of
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various properties of minimal clones are very important. We have studied a particular problem concerning minimal clones: Given a pair (CィイD21ィエD2, CィイD22ィエD2) of minimal clones, we call it gigantic pair if the union CィイD21ィエD2∪CィイD22ィエD2 generates the whole set of functions. We proved a characterization theorem of gigantic pairs and showed that gigantic pairs exist for most k's. 3. Study of hyperclones Recently, I. G. Rosenberg initiated the study of hyperclones. We continued his work and proved the following: The cardinality of the lattice of all hyperclones on the set {0,1} is of continuum. This is interesting as the cardinality of the lattice of all (ordinary) clones on {0, 1} is countable. 4. Study of partial clones consisting of partial functions We investigated the following problems on partial clones : (1) The minimal number of maximal partial clones whose meet is the trivial partial clone. (2) The minimal number of minimal partial clones whose join is the clone of all partial operations. This is a joint work with Professors L. Haddad and I.G. Rosenberg. Less
