Classification of minimal clones over a finite field in multiple-valued logic

多值逻辑中有限域上的最小克隆的分类

• 批准号: 20540111
• 负责人: MACHIDA Hajime
• 金额: $ 2.83万
• 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: 2008
• 资助国家: 日本
• 起止时间: 2008 至 2010
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-20540111/
• 关键词: 離散数学; 普遍代数; 多値論理

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.

对于固定集合a，定义在a上的多变量函数集，如果它包含所有的投影，并且对（泛函）复合是封闭的，则称为克隆。A上所有克隆的集合具有晶格结构。克隆晶格中的原子称为最小克隆。本文将有限域的结构引入基集a中，并将最小克隆的生成函数视为有限域a上的多项式。在此观点下，我们旨在对最小克隆进行分类。对于由二元幂等函数或三元多数函数生成的极小克隆，我们成功地找到了它们的极小函数在任意A下的素数幂的多项式表达式。

项目成果

Centralizing monoids, witnesses and minimal clones

集中幺半群、见证者和最小克隆

• 发表时间: 2010
• 作者: Fujita;Takahiko;松浦勉;Jun Sekine;H. Machida;H.Machida;Tsutomu Matsuura;H. Machida
• 通讯作者: H. Machida

Essentially minimal clones : Retrospective and prospective

本质上最小的克隆：回顾性和前瞻性

• 发表时间: 2009
• 作者: M. Watanabe and F. Kawai;H. Machida
• 通讯作者: H. Machida

On endoprimal monoids in clone theory

论克隆理论中的内源幺半群

• 发表时间: 2009
• 作者: H. Machida;I. G. Rosenberg
• 通讯作者: I. G. Rosenberg

Endoprimal monoids, commutation and witness

内源幺半群、换向和见证

• 发表时间: 2009
• 作者: M.Watanabe;F.Kawai;H.Machida;M. Watanabe and F. Kawai;小松和志;関根順;H. Machida
• 通讯作者: H. Machida

Some endoprimal monoids over a three-element set

三元素集上的一些内源幺半群

• 发表时间: 2009
• 作者: T. Delia Nova;H. Mawengkang;M. Watanabe;関根順;小松和志;H. Machida
• 通讯作者: H. Machida