GALOIS CONNEXIONS
GALOIS CONNEXIONS
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伽罗瓦联系
DOI：
10.1090/s0002-9947-1944-0010555-7
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发表时间：
2009
影响因子：
0.8
通讯作者：
Oystein Ore
中科院分区：
文献类型：
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作者：
Oystein Ore
本文包含在主要的一节我的学术讨论会讲座理论的数学关系在1941年夏季会议上的美国数学学会在芝加哥大学。（对这些讲座的简要回顾，见芝加哥夏季会议的报告，《美国数学学会公报》第48卷（1942年），第110页。169-182.）由于种种原因，我不得不推迟讨论会系列中关于这个问题的书，可能要等到战后。然而，我发现在目前发表这一理论的某些部分是可取的。作出这一决定的一个原因是，我曾在不同的时候与其他人讨论过这个理论的各个方面，他们对这些问题感兴趣，以至于希望发表自己的贡献。本文的目的是讨论一般类型的对应结构，我称之为伽罗瓦连接。这些对应出现在各种各样的数学理论中，并出现在关系论的几个例子中。因此，对它们的主要性质和解释进行单独的阐述似乎是可取的。这个名字取自普通伽罗瓦方程理论，其中子群和子域之间的对应表示这种类型的特殊对应。在对闭包关系作了一些介绍之后，讨论了伽罗瓦连接的一般性质。其次，它表明，每一个伽罗瓦连接可以设想为被定义的手段，一个连续的映射和相反，每一个映射的闭包关系定义一个伽罗瓦连接。通过二元关系可以给出不同的解释。加勒特·伯克霍夫已经指出，任何二元关系都定义了两个集合的子集之间的伽罗瓦连接类型的对应，并且很容易看出，相反地，每个伽罗瓦连接都可以以这种方式构造。作为一个说明性的例子，所有的二元关系与一个完美的伽罗瓦连接确定。伽罗瓦连接定义了一对对偶拓扑，使得这样的拓扑可以通过二元关系来定义。讨论了自对偶拓扑的构造。本文简要地指出了关系的一般伽罗瓦理论的可能性，并以等价关系的情形为例加以解决。最后，对置换群所定义的伽罗瓦连接作了一些说明。
This paper contains in the main a section of my Colloquium lectures on the theory of Mathematical relations given in 1941 at the Summer Meeting of the American Mathematical Society at the University of Chicago.(A brief review of these lectures can be found in the report of the Summer meeting at Chicago, Bull. Amer. Math. Soc. vol. 48 (1942) pp. 169-182.) Due to vari-ous causes it has been necessary for me to postpone the book on the subject in the Colloquium Series, probably until after the war. I have found it desirable, however, to publish certain parts of this theory at the present time. A contributing reason for this decision is the fact that I have at various times discussed aspects of the theory with others who have become interested in these problems to the extent of wishing to publish contributions of their own. The object of this paper is to discuss a general type of correspondence between structures which I have called Galois connexions. These correspondences occur in a great variety of mathematical theories and in several in-stances in the theory of relations. It seemed desirable therefore to give a sepa-rate exposition of their main properties and interpretations. The name is taken from the ordinary Galois theory of equations where the correspondence between subgroups and subfields represents a special correspondence of this type. After some introductory remarks on closure relations the general proper-ties of Galois connexions are discussed. Next it is shown that every Galois connexion can be conceived of as being defined by means of a continuous mapping and conversely every mapping of a closure relation defines a Galois connexion. A different interpretation can be given by means of binary relations. It has already been pointed out by Garrett Birkhoff that any binary relation defines a correspondence of the type of a Galois connexion between the subsets of two sets and it is easily seen that conversely every Galois con-nexion can be constructed in this manner. As an illustrative example all binary relations with a perfect Galois connexion are determined. The Galois connexion defines a pair of dual topologies so that such topologies can be defined by means of binary relations. The construction of self-dual topologies is discussed. The possibility of a general Galois theory for relations is indicated briefly and the case of an equivalence relation is solved as an example, There are some final remarks on the Galois connexion defined by a permutation group.