# 伽罗瓦连接

## 定义

F(a)<= b当且仅当aG(b)。

f (a)<= b当且仅当af (b)。

bF(a) 当且仅当aG(b)。

## 性质

f (f (f (x))) ≥ f (x)

f f f f (x) ≤ f f (x)≤ x,

f (f (f (x))) ≤ f (x)。

f (f (f (f (x)))) = f (f (x)),

## 注释

1. ^ 单调性可以从下面的条件得出。参见性质章节的讨论。明确出现在定义中是为了区别于可供选择的“反序”定义。你还可以定义伽罗瓦连接为满足如下松散条件的一对单调函数，对于所有A中x有x ≤ g(f (x))并且对于所有B中的y有f(g (y)) ≤ y。

## 引用

A freely available introduction to Galois connections, presenting many examples and results. Also includes notes on the different notations and definitions that arose in this area:

• M. Erné, J. Koslowski, A. Melton, G. E. Strecker, A primer on Galois connections, in: Proceedings of the 1991 Summer Conference on General Topology and Applications in Honor of Mary Ellen Rudin and Her Work, Annals of the New York Academy of Sciences, Vol. 704, 1993, pp. 103-125. Available online in various file formats: PS.GZ PS

The following standard reference books also include Galois connections using modern notation and definitions:

• B. A. Davey and H. A. Priestley: Introduction to lattices and Order, Cambridge University Press, 2002.
• G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. Mislove, D. S. Scott: Continuous Lattices and Domains, Cambridge University Press, 2003.

Finally, some publications using the original (antitone) definition:

• Garrett Birkhoff: Lattice Theory, Amer. Math. Soc. Coll. Pub., Vol 25, 1940
• Oystein Ore: Galois Connexions, Transactions of the American Mathematical Society 55 (1944), pp. 493-513