Galois connection in mathematical clone theory

数学克隆理论中的伽罗瓦联系

• 批准号: 13640106
• 负责人: MACHIDA Hajime
• 金额: $ 2.11万
• 依托单位: Hitotsubashi University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2001
• 资助国家: 日本
• 起止时间: 2001 至 2002
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640106/
• 关键词: (mathematical) clone; Galois connection; lattice of clones

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.

对于集合A，A上的克隆是一个在复合下封闭的多元函数集合。用L_A表示A上所有克隆的集合。本文对A上由一元函数构成的幺半群的集合M_A，考虑了M_A与L_A之间的一个自然定义的Galois联络。对于幺半群M，M的中心化子M^* 是所有与M中的每个一元函数“交换”的多元函数的集合。<Some fundamental properties of the Galois connection>（i）证明了么半群的所有中心化子都包含在某些特定的极大克隆中。(ii)同时，我们证明了对于每对不同的幺半群，它们的中心化子总是不同的。<Characterization of the centralizers of the symmetric group and the alternating group>:我们建立了对称群和交错群的中心化子的刻画，后者比前者更复杂. <Classification of the centralizers for a sequence of monoids which contain the symmetric group>定义了包含对称群的幺半群的典型序列{N_i}。确定了所有N_i的中心化子。大多数与最少的克隆相吻合. <Monoids whose centralizer is the least done>：认为在伽罗瓦联系下，一个小幺半群对应于一个大幺半群是“自然的”。然而，与这种天真的直觉相反，已经发现了一些小幺半群，它们的中心化子是最小的克隆。<Application of the Kuznetsov criterion>库兹涅佐夫建立的标准的力量在我们的研究中被证明是非常有用的。

项目成果

H.Machida: "Hyperclones on a two-element set"Multiple-Valued Logic -An International Journal -. 8. 495-501 (2002)

H.Machida：“二元集上的超克隆”多值逻辑 - 国际期刊 -。

H.Machida: "Some results on the centralizers of monoids in clone theory"Proceedings 32nd International Symposium on Multiple-Valued Logic. 10-16 (2002)

H.Machida：“克隆理论中幺半群集中化的一些结果”第 32 届国际多值逻辑研讨会论文集。

Haddad, L., Machida, H. and Rosenberg, I. G.: "Maximal and minimal partial clones"Journal of Automata, Languages and Combinatorics. 7. 83-93 (2002)

Haddad, L.、Machida, H. 和 Rosenberg, I. G.：“最大和最小部分克隆”自动机、语言和组合学杂志。

I.G.Rosenberg: "Gigantic pairs of minimal clones-Characterization and existence"Multiple-Valued Logic-An International Journal. Vol.7. 129-148 (2001)

I.G.Rosenberg：“最小克隆的巨大对 - 特征和存在”多值逻辑 - 国际期刊。

H.Machida: "Relations between clones and full monoids"Proceedings 31st International Symposium on Multiple-Valued Logic. 279-284 (2001)

H.Machida：“克隆与完整幺半群之间的关系”第 31 届国际多值逻辑研讨会论文集。