PlanetMath: Galois connection

(more info)

Math for the people, by the people.

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (210)
Orphanage
Unclass'd (1)
Unproven (472)
Corrections (38)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Legalese
About

Entry average rating: No information on entry rating

Galois connection

(Definition)

The notion of a Galois connection has its root in Galois theory. By the fundamental theorem of Galois theory, there is a one-to-one correspondence between the intermediate fields between a field and its subfield (with appropriate conditions imposed on the extension ), and the subgroups of the Galois group $\operatorname{Gal}(L/F)$ such that the bijection is inclusion-reversing:

$\displaystyle \operatorname{Gal}(L/F)\supseteq H\supseteq\langle e \rangle$ iff $\displaystyle \quad F\subseteq L^H\subseteq L,$ and

$\displaystyle F\subseteq K\subseteq L$ iff $\displaystyle \quad \operatorname{Gal}(L/F)\supseteq \operatorname{Gal}(L/K)\supseteq\langle e \rangle.$

If the language of Galois theory is distilled from the above paragraph, what remains reduces to a more basic and general concept in the theory of ordered-sets:

Definition. Let $(P, \le_P)$ and $(Q, \le_Q)$ be two posets. A Galois connection between $(P,\le_P)$ and $(Q,\le_Q)$ is a pair of functions with $f^*\colon P\to Q$ and $f_*\colon Q\to P$ , such that, for all $p\in P$ and $q\in Q$ , we have

$\displaystyle f^*(p)\leq_Q q$ iff $\displaystyle \quad p\leq_P f_*(q).$

We denote a Galois connection between

and

by $P\stackrel{f}{\multimap}Q$ , or simply $P\multimap Q$ .

If we define $\le_P^{\prime}$ on

by $a\le_P^{\prime}b$ iff $b\le_P a$ , and define $\le_Q^{\prime}$ on

by $c\le_P^{\prime}d$ iff $c\le_P d$ , then $(P,\le_P^{\prime})$ and $(Q,\le_Q^{\prime})$ are posets, (the duals of $(P,\le_P)$ and $(Q,\le_Q)$ ). The existence of a Galois connection between $(P,\le_P)$ and $(Q,\le_Q)$ is the same as the existence of a Galois connection between $(Q,\le_Q^{\prime})$ and $(P,\le_P^{\prime})$ . In short, we say that there is a Galois connection between

and

if there is a Galois connection between two posets

and

where

and

are the underlying sets (of

and

respectively). With this, we may say without confusion that “a Galois connection exists between

and

iff a Galois connection exists between

and

”.

Remarks.

Since $f^*(p)\leq_Q f^*(p)$ for all $p\in P$ , then by definition, $p\leq_P f_*f^*(p)$ . Alternatively, we can write

$\displaystyle 1_P\leq_P f_*f^*,$ (1)

where stands for the identity map on . Similarly, if is the identity map on , then

$\displaystyle f^*f_*\leq_Q 1_Q.$ (2)
Suppose $a\leq_P b$ . Since $b\leq_P f_*f^*(b)$ by the remark above, $a\leq_P f_*f^*(b)$ and so by definition, $f^*(a)\leq_Q f^*(b)$ . This shows that is monotone. Likewise, is also monotone.
Now back to Inequality (1), $1_P\leq_P f_*f^*$ in the first remark. Applying the second remark, we obtain

$\displaystyle f^*\leq_Q f^*f_*f^*.$ (3)

Next, according to Inequality (2), $f^*f_*(q)\leq_Q q$ for any $q\in Q$ , it is true, in particular, when . Therefore, we also have

$\displaystyle f^*f_*f^*\leq_Q f^*.$ (4)

Putting Inequalities (3) and (4) together we have

$\displaystyle f^*f_*f^*=f^*.$ (5)

Similarly,

$\displaystyle f_*f^*f_*=f_*.$ (6)

Examples.

The most famous example is already mentioned in the first paragraph above: let is a finite-dimensional Galois extension of a field , and $G:=\operatorname{Gal}(L/F)$ is the Galois group of over . If we define

a.

$P=\lbrace K\mid K$ is a field such that $F\subseteq K\subseteq L \rbrace,$ with $\leq_P=\subseteq$ ,

b.

$Q=\lbrace H\mid H$ is a subgroup of $G \rbrace,$ with $\leq_Q=\supseteq$ ,

c.

$f^*:P\to Q$ by $f^*(K)=\operatorname{Gal}(L/K)$ , and

d.

$f_*:Q\to P$ by , the fixed field of in .

Then, by the fundamental theorem of Galois theory, and are bijections, and is a Galois connection between and .
Let be a topological space. Define be the set of all open subsets of and the set of all closed subsets of . Turn and into posets with the usual set-theoretic inclusion. Next, define $f^*:P\to Q$ by $f^*(U)=\overline{U}$ , the closure of , and $f_*:Q\to P$ by $f_*(V)=\operatorname{int}(V)$ , the interior of . Then is a Galois connection between and . Incidentally, those elements fixed by are precisely the regular open sets of , and those fixed by are the regular closed sets.

"Galois connection" is owned by CWoo. [ full author list (2) ]

(view preamble)