2002 Fiscal Year Final Research Report Summary
Galois connection in mathematical clone theory
| Project/Area Number |
13640106
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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. Grad. S., Commerce and Management, Professor, 大学院・商学研究科, 教授 (40090534)
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| Co-Investigator(Kenkyū-buntansha) |
YAMADA Hiromichi Hitotsubashi Univ. Grad. S., Economics, Professor, 大学院・経済学研究科, 教授 (50134888)
YAMASAKI Hideki Hitotsubashi Univ. Grad. S., Commerce and Management, Professor, 大学院・商学研究科, 教授 (30108188)
IWASAKI Shiro Hitotsubashi Univ. Grad. S., Economics, Professor, 大学院・経済学研究科, 教授 (00001842)
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| Project Period (FY) |
2001 – 2002
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| Keywords | (mathematical) clone / Galois connection / lattice of clones |
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| Research Abstract |
For a set A, a clone on A is a set of multi-variable functions which is closed under composition. Denote by L_A the set of all clones on A. In this reseach, for the set M_A of all monoids consisting of unary functions on A, we considered a naturally defined Galois connection between M_A and L_A. For a monoid M, the centralizer M^* of M is the set of all multi-variable functions which 'commutes' with every unary function in M.
1. <Some fundamental properties of the Galois connection> : (i) We showed that all centralizers of monoids are contained in some particular maximal clones. (ii) Also, we showed that for every pair of distinct monoids their centralizers are always distinct.
2. <Characterization of the centralizers of the symmetric group and the alternating group> : We established the characterization of the centralizers of both the symmetric group and the alternating group, the latter being more complex than the former.
3. <Classification of the centralizers for a sequence of monoids which contain the symmetric group> : A typical sequence {N_i} of monoids containing the symmetric group was defined. The centralizers of all N_i's have been determined. Most of them coincide with the least clone.
4. <Monoids whose centralizer is the least done> : It is 'natural' to think that under a Galois connection a small monoid corresponds to a big monoid. However, against this naive intuition, some small monoids have been discovered whose centralizer is the least clone.
5. <Application of the Kuznetsov criterion> : The power of the criterion established by Kuznetsov was shown to be quite useful in our study.
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