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上的多变量函数的集合，它在复合下是封闭的。A上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一元群的格，在本研究中，我们考虑了两个格之间的自然定义的伽罗瓦连接。对于单oid M， M的中心化器是与M.1中所有一元函数“交换”的所有多变量函数的集合。伽罗瓦连接的几个基本性质：(1)我们粗略地研究了单调群的中心中心在克隆格中的位置。(2)我们证明了对于每一对不同的单群，它们的中心点总是不同的。2 .对称群和交替群的中心子的性质：建立了对称群和交替群的中心子的性质。单似群扶正器最小克隆的一个充分条件：我们找到了单似群扶正器最小克隆的一个充分条件，可以作为一个非常方便的工具来使用。包含对称群的一元群的中心子：我们确定了所有包含对称群的一元群的中心子。在本研究的过程中，上述充分条件得到了相当有效的利用。对于大多数monoids来说，它们的集中器被证明是最少克隆的。然而，在4个元素集合上定义的单群“M_2”是一个例外，它的扶正器不是最不克隆的。正集是最小克隆的一元群：我们很自然地认为，在伽罗瓦连接下，一个小一元群对应一个大一元群。然而，与这种直觉相反的是，已经发现了一些小的一元群，它们的扶正器是最少克隆的。

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.