2013 Fiscal Year Final Research Report
Theory of commutation and minimal clones in multiple-valued logic
| Project/Area Number |
23540158
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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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 | International Christian University |
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Principal Investigator |
MACHIDA Hajime 国際基督教大学, アーツ・サイエンス研究科, 研究員 (40090534)
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| Project Period (FY) |
2011 – 2013
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| Keywords | 多値論理 / 普遍代数 / 離散数学 |
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| Research Abstract |
A set of multi-variable functions defined on a given set A is a clone on A if it is closed under functional composition and contains all projections on A. The set of clones on A forms a lattice. Except the case for |A|=2, the structure of the clone lattice on A is extremely complex and, until now, mostly unknown.
Based on the commutativity of functions the notion of a centralizer (centralizing clone) is defined and, furthermore, a centralizing monoid is defined as the set of unary functions of a centralizer. In this project, for the case of |A|=3, we determined all centralizing monoids on A as well as the inclusion relations among them. We also investigated the relation between maximal centralizing monoids and majority functions which generate minimal clones.
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