Studies on the structure of the lattice of clones consisting of multiple-valued logical functions

多值逻辑函数克隆格结构研究

• 批准号: 10640109
• 负责人: MACHIDA Hajime
• 金额: $ 1.41万
• 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: 1998
• 资助国家: 日本
• 起止时间: 1998 至 1999
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-10640109/
• 关键词: multiple-valued logical function; (mathematical) clone; lattice of clones; minimal clone; hyperclone; クローン理論

A clone is a set of k-valued logical functions which is closed under composition and contains all the projections. The set of all clones consisting of k-valued logical functions is denoted by LィイD2kィエD2. Whereas the structure of LィイD22ィエD2 is completely determined, our knowledge about the structure of LィイD2kィエD2 for k > 2 at present is very little. The main objective of this research is to clarify the structure of LィイD2kィエD2 and we have obtained the following results.1. The structure of LィイD23ィエD2 as a metric spaceWe have introduced a metric into the lattice LィイD2kィエD2 of clones and showed that LィイD2kィエD2 is a compact metric space. Moreover, we constructed continuous mappings, based on the meet operation, from LィイD23ィエD2 onto LィイD22ィエD2 and studied the images of maximal clones in LィイD23ィエD2 and those of some clones being accumulation points under such mappings.2. Minimal clones in LィイD2kィエD2 and related topicsSince the classification of minimal clones is far from complete, the study of … More various properties of minimal clones are very important. We have studied a particular problem concerning minimal clones: Given a pair (CィイD21ィエD2, CィイD22ィエD2) of minimal clones, we call it gigantic pair if the union CィイD21ィエD2∪CィイD22ィエD2 generates the whole set of functions. We proved a characterization theorem of gigantic pairs and showed that gigantic pairs exist for most k's.3. Study of hyperclonesRecently, I. G. Rosenberg initiated the study of hyperclones. We continued his work and proved the following: The cardinality of the lattice of all hyperclones on the set {0,1} is of continuum. This is interesting as the cardinality of the lattice of all (ordinary) clones on {0, 1} is countable.4. Study of partial clones consisting of partial functionsWe investigated the following problems on partial clones : (1) The minimal number of maximal partial clones whose meet is the trivial partial clone. (2) The minimal number of minimal partial clones whose join is the clone of all partial operations. This is a joint work with Professors L. Haddad and I.G. Rosenberg. Less

克隆是k值逻辑函数的集合，它在复合下是封闭的，并且包含所有的投影。由k值逻辑函数组成的所有克隆的集合用L ` ` ` ` ` ` ` ` ` ` ` ` D2表示。虽然L的结构是完全确定的，但我们目前对L的结构知之甚少。本研究的主要目的是阐明L ` ` ` D2k ` ` D2的结构，我们得到了以下结果：我们在无克隆的晶格中引入了一个度量，并证明了L ` ` D2k ` ` D2是一个紧化度量空间。此外，我们基于满足运算构造了从L′′′D23′′的连续映射到L′′D22′的连续映射，并研究了L′′D23′的连续映射中最大克隆的图像以及在该映射下作为累加点的一些克隆的图像。由于最小克隆的分类还很不完整，因此对最小克隆的更多性质的研究是非常重要的。我们研究了一个关于最小克隆的特殊问题：给定一个最小克隆对（C D21 D2, C D22 D2），如果C D21 D2∪C D22 D2生成整个函数集，我们称其为巨大对。我们证明了巨大对的一个表征定理，并证明了对大多数k'存在巨大对。最近，I. G. Rosenberg开始了对超克隆的研究。我们继续他的工作，证明了{0,1}上所有超克隆的格的基数是连续体的。这很有趣，因为{0,1}上所有（普通）克隆的格的基数是可数的。由部分函数组成的部分无性系的研究我们研究了部分无性系的下列问题：(1)满足平凡部分无性系的最大部分无性系的最小数目。(2)最小部分克隆的最小数目，其连接是所有部分操作的克隆。这是与L. Haddad教授和I.G. Rosenberg教授的合作成果。少

项目成果

MACHIDA, Hajime: "Hyperclones on the two-element set"Multiple-Valued Logic - An International Journal. (発表予定).

MACHIDA, Hajime：“二元集的超克隆”多值逻辑 - 国际期刊（即将出版）。

KOZONO,H.: "Exterior problem for the stationary Navier-Stokes equations in the Lorents space" Math.Ann.318. 279-305 (1998)

KOZONO,H.：“洛伦兹空间中平稳纳维-斯托克斯方程的外部问题”Math.Ann.318。

MACHIDA, Hajime: "Prelude to local complexity theory"数理解析研究所講究録(京都大学). 1054. 87-95 (1998)

MACHIDA, Hajime：“局部复杂性理论的前奏”数学分析研究所的 Kokyuroku（京都大学）1054. 87-95 (1998)。

MACHIDA,Hajime: "Some continuous maps on the space of clones in multiple-valued logic" Proc.of 28th International Symp.on Multiple-Valued Logic. 28. 374-379 (1998)

MACHIDA,Hajime：“多值逻辑中克隆空间的一些连续映射”Proc.of 28th International Symp.on Multiple-Valued Logic。

KITAZUME,M.: "Ternary codes and vertex operator algebras" Journal of Algebra. (発表予定).

KITAZUME, M.：“三元码和顶点算子代数”代数杂志（待出版）。