2007 Fiscal Year Final Research Report Summary
Classification of minimal clones in multiple-valued logic and finite fields
| Project/Area Number |
18540116
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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 University, Grad. S. of Commerce and Management, Professor (40090534)
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| Co-Investigator(Kenkyū-buntansha) |
IWASAKI Shiro Hitotsubashi University, Emeritus Professor (00001842)
YAMASAKI Hideki Hitotsubashi University, Research and Development Center for Higher Educationf, Professor (30108188)
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| Project Period (FY) |
2006 – 2007
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| Keywords | multiple-valued logic / universal algebra / clone theory / minimal clone |
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| Research Abstract |
For a set A, a clone on A is a set of multi-variable functions on A which contains all projections and is closed under composition The set of all clones on A forms a lattice which is called the clone lattice over A and is denoted by L_A. The structure of L_A is extremely complex and , up to now, largely unknown.
The clone J_A of all projections is clearly the least element of the lattice L_A. A minimal element of the set L_A-J A is called a minimal clone. A minimal clone is generated by a single function, which is called a minimal function..
In this research, we employed a new method of research by introducing the structure of a finite field GF(|A|) into the finite base set A and treating minimal functions over the set A as polynomials defined over GF(|A|). In this way we tried to grasp the properties of minimal functions as the properties of polynomials over a finite field. Starting from the result of B. Csakany who presented all minimal clones over a three-element set, we first expressed Csakany's minimal functions as polynomials over GF(3) and generalized some of them to polynomials over arbitrary finite field GF(k) where k is any prime power.
More specifically, we considered the cases where minimal functions are (1) binary idempotent functions and (2) ternary majority functions. We obtained, among others, the complete characterization of binary linear minimal polynomials as well as binary minimal monomials. Furthermore, we have obtained many examples of polynomials generating minimal clones over GF(k) for any prime power k.
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