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

对于集合A，A上的克隆是A上在合成下闭合的多元函数的集合。A上所有克隆的集合形成了格。我们还考虑了由一元函数组成的所有么半群的格，在本研究中，我们考虑了这两个格之间的自然定义的Galois联系。对于么半群M，M的中心化子是与M1中所有一元函数‘交换’的所有多元函数的集合。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)证明了对于每一对不同的么半群，它们的中心化子总是不同的。2.对称群和交错群的中心化子的特征：我们建立了对称群和交错群的中心化子的刻画。3.么半群的中心化子是最小克隆的一个充分条件：我们找到了么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.包含对称群的么半群的中心化子：我们确定了所有包含对称群的么半群的中心化子。在本研究过程中，上述充分条件得到了相当有效的应用。对于大多数么半群来说，它们的中间体是最小的克隆。然而，定义在4元集上的么半群“M_2”是一个例外，它的中心子不是最小的克隆。5.中心子是最小克隆的么半群：人们很自然地认为在伽罗瓦联络下，一个小么半群对应一个大么半群。然而，与这一直觉相反的是，人们发现了一些小么半群，它们的中心线是最小的克隆。

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.