1999 Fiscal Year Final Research Report Summary
Studies on the structure of the lattice of clones consisting of multiple-valued logical functions
| Project/Area Number |
10640109
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | Hitotsubashi University |
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Principal Investigator |
MACHIDA Hajime Hitotsubashi Univ. Dept. of Commerce, Professor, 商学部, 教授 (40090534)
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| Co-Investigator(Kenkyū-buntansha) |
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)
IWASAKI Shiro Grad. S., Economics, Professor, 大学院・経済学研究科, 教授 (00001842)
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)
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| Project Period (FY) |
1998 – 1999
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| Keywords | multiple-valued logical function / (mathematical) clone / lattice of clones / minimal clone / hyperclone |
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| Research Abstract |
A clone is a set of k-valued logical functions which is closed under composition and contains all the projections. The set of all clones consisting of k-valued 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
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