Galois connection

Galois connection

The notion of a

Galois connection

has its root in

Galois theory

. By the

fundamental theorem of Galois theory

(

http://planetmath.org/FundamentalTheoremOfGaloisTheory

), there is a

one-to-one correspondence

between the intermediate fields between a field

L

and its

subfield

F

(with appropriate conditions imposed on the

extension

L

/

F

), and the subgroups of the

Galois group

Gal

⁡

(

L

/

F

)

such that the bijection is inclusion-reversing:

Gal

⁡

(

L

/

F

)

⊇

H

⊇

⟨

e

⟩

iff

F

⊆

L

H

⊆

L

,

and

F

⊆

K

⊆

L

iff

Gal

⁡

(

L

/

F

)

⊇

Gal

⁡

(

L

/

K

)

⊇

⟨

e

⟩

.

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

,

≤

P

)

and

(

Q

,

≤

Q

)

be two posets. A

Galois connection

between

(

P

,

≤

P

)

and

(

Q

,

≤

Q

)

is a pair of

functions

f

:=

(

f

*

,

f

*

)

with

f

*

:

P

→

Q

and

f

*

:

Q

→

P

, such that, for all

p

∈

P

and

q

∈

Q

, we have

f

*

⁢

(

p

)

≤

Q

q

iff

p

≤

P

f

*

⁢

(

q

)

.

We denote a Galois
connection between

P

and

Q

by

P

⊸

f

Q

, or simply

P

⊸

Q

.

If we define

≤

P

′

on

P

by

a

≤

P

′

b

iff

b

≤

P

a

, and define

≤

Q

′

on

Q

by

c

≤

Q

′

d

iff

d

≤

Q

c

, then

(

P

,

≤

P

′

)

and

(

Q

,

≤

Q

′

)

are posets, (the duals of

(

P

,

≤

P

)

and

(

Q

,

≤

Q

)

). The

existence

of a Galois connection between

(

P

,

≤

P

)

and

(

Q

,

≤

Q

)

is the same as the existence of a Galois connection between

(

Q

,

≤

Q

′

)

and

(

P

,

≤

P

′

)

. In short, we say that there is a Galois connection between

P

and

Q

if there is a Galois connection between two posets

S

and

T

where

P

and

Q

are the underlying sets (of

S

and

T

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

P

and

Q

iff a Galois connection exists between

Q

and

P

”.

Remarks.

1.

Since

f

*

⁢

(

p

)

≤

Q

f

*

⁢

(

p

)

for all

p

∈

P

, then by definition,

p

≤

P

f

*

⁢

f

*

⁢

(

p

)

. Alternatively, we can write

1

P

≤

P

f

*

⁢

f

*

,

(1)

where

1

P

stands for the

identity map

on

P

. Similarly, if

1

Q

is the identity map on

Q

, then

f

*

⁢

f

*

≤

Q

1

Q

.

(2)

2.

Suppose

a

≤

P

b

. Since

b

≤

P

f

*

⁢

f

*

⁢

(

b

)

by the remark above,

a

≤

P

f

*

⁢

f

*

⁢

(

b

)

and so by definition,

f

*

⁢

(

a

)

≤

Q

f

*

⁢

(

b

)

. This shows that

f

*

is

monotone

. Likewise,

f

*

is also monotone.

3.

Now back to

Inequality

(1),

1

P

≤

P

f

*

⁢

f

*

in the first remark. Applying the second remark, we obtain

f

*

≤

Q

f

*

⁢

f

*

⁢

f

*

.

(3)

Next, according to Inequality (2),

f

*

⁢

f

*

⁢

(

q

)

≤

Q

q

for any

q

∈

Q

, it is true, in particular, when

q

=

f

*

⁢

(

p

)

. Therefore, we also have

f

*

⁢

f

*

⁢

f

*

≤

Q

f

*

.

(4)

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

f

*

⁢

f

*

⁢

f

*

=

f

*

.

(5)

Similarly,

f

*

⁢

f

*

⁢

f

*

=

f

*

.

(6)

4.

If

(

f

,

g

)

and

(

f

,

h

)

are Galois connections between

(

P

,

≤

P

)

and

(

Q

,

≤

Q

)

, then

g

=

h

. To see this, observe that

p

≤

P

g

⁢

(

q

)

iff

f

⁢

(

p

)

≤

Q

q

iff

p

≤

P

h

⁢

(

q

)

, for any

p

∈

P

and

q

∈

Q

. In particular, setting

p

=

g

⁢

(

q

)

, we get

g

⁢

(

q

)

≤

P

h

⁢

(

q

)

since

g

⁢

(

q

)

≤

P

g

⁢

(

q

)

. Similarly,

h

⁢

(

q

)

≤

P

g

⁢

(

q

)

, and therefore

g

=

h

. By a similarly

argument

, if

(

g

,

f

)

and

(

h

,

f

)

are Galois connections between

(

P

,

≤

P

)

and

(

Q

,

≤

Q

)

, then

g

=

h

. Because of this uniqueness property, in a Galois connection

f

=

(

f

*

,

f

*

)

,

f

*

is called

the upper adjoint

of

f

*

and

f

*

the lower adjoint

of

f

*

.

Examples.

•

The most famous example is already mentioned in the first paragraph above: let

L

is a finite-dimensional

Galois extension

of a field

F

, and

G

:=

Gal

⁡

(

L

/

F

)

is the Galois group of

L

over

F

. If we define

a.

P

=

{

K

∣

K

⁢

is a field such that

⁢

F

⊆

K

⊆

L

}

,

with

≤

P

⁣

=

⁣

⊆

,

b.

Q

=

{

H

∣

H

⁢

is a subgroup of

⁢

G

}

,

with

≤

Q

⁣

=

⁣

⊇

,

c.

f

*

:

P

→

Q

by

f

*

⁢

(

K

)

=

Gal

⁡

(

L

/

K

)

, and

d.

f

*

:

Q

→

P

by

f

*

⁢

(

H

)

=

L

H

, the

fixed field

of

H

in

L

.

Then, by the fundamental theorem of Galois theory,

f

*

and

f

*

are bijections, and

(

f

*

,

f

*

)

is a Galois connection between

P

and

Q

.

•

Let

X

be a

topological space

. Define

P

be the set of all

open subsets

of

X

and

Q

the set of all

closed subsets

of

X

. Turn

P

and

Q

into posets with the usual set-theoretic

inclusion

. Next, define

f

*

:

P

→

Q

by

f

*

⁢

(

U

)

=

U

¯

, the

closure

of

U

, and

f

*

:

Q

→

P

by

f

*

⁢

(

V

)

=

int

⁡

(

V

)

, the

interior

of

V

. Then

(

f

*

,

f

*

)

is a Galois connection between

P

and

Q

. Incidentally, those

elements

fixed by

f

*

⁢

f

*

are precisely the

regular open sets

of

X

, and those fixed by

f

*

⁢

f

*

are the regular closed sets.

Remark

. The pair of functions in a Galois connection are order preserving as shown above. One may also define a Galois connection as a pair of maps

f

*

:

P

→

Q

and

f

*

:

Q

→

P

such that

f

*

⁢

(

p

)

≤

Q

q

iff

f

*

⁢

(

q

)

≤

P

p

, so that the pair

f

*

,

f

*

are order reversing. In any case, the two definitions are

equivalent

in that one may go from one definition to another, (simply exchange

Q

with

Q

∂

, the dual (

http://planetmath.org/DualPoset

) of

Q

).

References

1

T.S. Blyth,

Lattices and Ordered

Algebraic Structures

, Springer, New York (2005).

2

B. A. Davey, H. A. Priestley,

Introduction

to Lattices and Order

, 2nd Edition, Cambridge (2003)

Title

Galois connection

Canonical name

GaloisConnection

Date of creation

2013-03-22 15:08:15

Last modified on

2013-03-22 15:08:15

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

13

Author

CWoo (3771)

Entry type

Definition

Classification

msc 06A15

Synonym

Galois correspondence

Synonym

Galois connexion

Related topic

InteriorAxioms

Related topic

AdjointFunctor

Defines

upper adjoint

Defines

lower adjoint

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