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The structure of the clone lattice and Galois connection in multiple-valued logic

多值逻辑中克隆格的结构和伽罗瓦连接

基本信息

批准号：
15540112
负责人：
MACHIDA Hajime
金额：
$ 2.24万
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：
2003
资助国家：
日本
起止时间：
2003 至 2004
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-15540112/
关键词：
(mathematical)clone lattice of clones Galois connection centralizer of a clone hyperclone

项目摘要

For a set A, a clone on A is a set of multi-variable functions on A which is closed under composition. The set of all clones on A forms the lattice. We also consider the lattice of all monoids consisting of unary functions, In this research, we considered a naturally defined Galois connection between both lattices. For a monoid M, the centralizer of M is the set of all multi-variable functions which ‘commutes' with all unary functions in M.1.Some fundamental properties of the Galois connection :(1)We studied roughly the positions of the centralizers of monoids in the lattice of clones. (2)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.3.A sufficient condition for the centralizer of a monoid to be the least clone :We found a sufficient condition for the centralizer of a monoid to be the least clone which can be used as a very convenient tool.4.The centralizers of monoids containing the symmetric group :We determined the centralizers of all monoids which contain the symmetric group. In the course of this research, the above mentioned sufficient condition has been used quite effectively. For most monoids, their centralizers turned out to be the least clone. However, the monoid "M_2" defined over the 4 element set is an exception and its centralizer is not the least clone.5.Monoids whose centralizer is the least clone :It is ‘natural' to think that under a Galois connection a small monoid corresponds to a large monoid. However, against this intuition, some small monoids have been discovered whose centralizer is the least clone.

对于集合A，A上的克隆是A上的一组多元函数，它在复合下是封闭的。A上的所有克隆的集合形成晶格。我们还考虑了由一元函数组成的所有幺半群的格，在这项研究中，我们考虑了两个格之间的一个自然定义的伽罗瓦连接。对么半群M，M的中心化子是与M中所有一元函数“可换”的所有多元函数的集合。1. Galois联络的一些基本性质：（1）我们粗略地研究了么半群的中心化子在克隆格中的位置。(2)We 2.对称群和交错群的中心化子的刻画：建立了对称群和交错群的中心化子的刻画3.幺半群的中心化子是最小克隆的充分条件：我们找到了幺半群的中心化子是最小克隆的充分条件，这是一个非常方便的工具4.包含对称群的幺半群的中心化子的刻画：我们确定了所有包含对称群的幺半群的中心化子。在研究过程中，上述充分条件得到了有效的应用。对于大多数幺半群来说，它们的中心化者是最小的克隆体。然而，定义在4元集上的幺半群“M_2”是一个例外，它的中心化子不是最小克隆。5.中心化子是最小克隆的幺半群：认为在伽罗瓦联络下，一个小幺半群对应于一个大幺半群是“自然的”。然而，与这种直觉相反，已经发现了一些小幺半群，其中心化子是最小克隆。