2010 Fiscal Year Final Research Report
Classification of minimal clones over a finite field in multiple-valued logic
| Project/Area Number |
20540111
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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, 大学院・アーツ・サイエンス研究科, 研究員 (40090534)
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| Co-Investigator(Kenkyū-buntansha) |
YAMASAKI Hideki 一橋大学, 大学院・商学研究科, 教授 (30108188)
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| Co-Investigator(Renkei-kenkyūsha) |
POGOSYAN Grant 国際基督教大学, 教養学部, 教授 (90234640)
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| Project Period (FY) |
2008 – 2010
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| Keywords | 離散数学 / 普遍代数 / 多値論理 |
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| 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.
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