congruence lattice

congruence lattice

Theorem 1

.

The set

Con

⁡

(

A

)

of all

congruences

on an

algebraic system

A

is a

complete lattice

that is a

sublattice

(called the

congruence lattice

of

A

) of the

lattice of equivalence relations

on

A

.

Proof.

It is not hard to see that

Con

⁡

(

A

)

is a

topped intersection structure

. As a result,

Con

⁡

(

A

)

is a complete lattice. Since it is also a subset of the

lattice

E

⁢

(

A

)

of

equivalence relations

on

A

, the only remaining item to show is that it is a sublattice of

E

⁢

(

A

)

. In other words, we need to show that if

𝒞

is a family of

congruence relations

on

A

, so is

Θ

:=

⋁

𝒞

.

For convenience, let us introduce some notational devices:

•

𝐧

:=

{

1

,

…

,

n

}

;

•

for any

a

k

,

b

k

∈

A

and

Θ

k

∈

Con

⁡

(

A

)

, where

k

∈

𝐧

, we mean

(

a

1

,

…

,

a

n

)

≡

(

b

1

,

…

,

b

n

)

(

mod

(

Θ

1

,

…

,

Θ

n

)

)

by

a

k

≡

b

k

(

mod

Θ

k

)

, for each

k

∈

𝐧

;

•

(

a

1

,

…

,

a

n

)

≡

(

b

1

,

…

,

b

n

)

(

mod

Θ

)

means

a

k

≡

b

k

(

mod

Θ

)

, for each

k

∈

𝐧

;

•

Finally, when

c

k

≡

c

k

+

1

(

mod

Θ

k

)

, where

k

∈

𝐧

-

𝟏

, we write

c

1

≡

Θ

1

c

2

≡

Θ

2

⋯

≡

Θ

n

-

1

c

n

.

Let us call this an (of

n

).

Back to the proof. Let

ω

be an

n

-ary

operator

on

A

and

(

a

1

,

…

,

a

n

)

≡

(

b

1

,

…

,

b

n

)

(

mod

Θ

)

. We want to show that

ω

⁢

(

a

1

,

…

,

a

n

)

≡

ω

⁢

(

b

1

,

…

,

b

n

)

(

mod

Θ

)

.

The proof now breaks up into two main steps:

Step 1

.

For each

i

∈

n

,

a

i

≡

b

i

(

mod

Θ

)

means we have a finite

a

i

=

c

1

≡

F

1

c

2

≡

F

2

⋯

≡

F

p

-

1

c

p

=

b

i

,

where each

F

i

is a congruence in

C

, and each

c

i

∈

A

. Now the lengths of the

sequences

may vary by

i

. The idea is to show that we can in fact pick the sequences so that they all have the same length.

To see this, take the first two pairs of

congruent

elements

a

1

≡

b

1

(

mod

Θ

)

and

a

2

≡

b

2

(

mod

Θ

)

, and expand them into two finite

1.

a

1

=

c

1

≡

F

1

c

2

≡

F

2

⋯

≡

F

p

-

1

c

p

=

b

1

, and

2.

a

2

=

d

1

≡

G

1

d

2

≡

G

2

⋯

≡

G

q

-

1

d

q

=

b

2

,

where

c

i

,

d

j

∈

A

and

F

i

,

G

j

∈

𝒞

. If

q

<

p

, we may lengthen the second chain so it has the same length as the first one:

a

2

=

d

1

≡

G

1

d

2

⁢

⋯

≡

G

q

-

1

d

q

≡

G

q

d

q

+

1

≡

G

q

+

1

⋯

≡

G

p

-

1

d

p

,

where

G

q

-

1

=

G

q

=

⋯

=

G

p

-

1

and

d

q

=

⋯

=

d

p

=

b

.

The above

argument

and an

induction

step on

n

show that we can indeed make all the “expanded” the same length. As a result, without loss of generality, we assume that all the expanded chains have the same length, say

r

+

1

.

Step 2

.

Complete

the proof.

Instead of writing all

n

chains, let us use our notational device, and we have the following

single

(again, we may write it in this fashion because all chains are now assumed to have the same finite length of

r

+

1

):

\xymatrix

⁢

(

c

11

,

…

,

c

1

⁢

n

)

⁢

\ar

⁢

@

⁢

3

-

[

r

⁢

r

]

-

⁢

(

F

11

,

…

,

F

1

⁢

n

)

⁢

&

⁢

&

⁢

(

c

21

,

…

,

c

2

⁢

n

)

⁢

\ar

⁢

@

⁢

3

-

[

r

⁢

r

]

-

⁢

(

F

21

,

…

,

F

2

⁢

n

)

⁢

&

⁢

&

⁢

⋯

⁢

\ar

⁢

@

⁢

3

-

[

r

⁢

r

]

-

⁢

(

F

r

⁢

1

,

…

,

F

r

⁢

n

)

⁢

&

⁢

&

⁢

(

c

r

+

1

,

1

,

…

,

c

r

+

1

,

n

)

where each

c

i

⁢

j

∈

A

, each

F

i

⁢

j

∈

𝒞

, with

(

i

,

j

)

∈

(

𝐫

+

𝟏

)

×

𝐧

, and that

(

a

1

,

…

,

a

n

)

=

(

c

11

,

…

,

c

1

⁢

n

)

and

(

c

r

+

1

,

1

,

…

,

c

r

+

1

,

n

)

=

(

b

1

,

…

,

b

n

)

.

Look at the first congruence pair

\xymatrix

⁢

(

c

11

,

…

,

c

1

⁢

n

)

⁢

\ar

⁢

@

⁢

3

-

[

r

⁢

r

]

-

⁢

(

F

11

,

…

,

F

1

⁢

n

)

⁢

&

⁢

&

⁢

(

c

21

,

…

,

c

2

⁢

n

)

. This can be expanded into an of length

n

as follows:

\xymatrix

⁢

(

c

11

,

c

12

,

…

,

c

1

⁢

n

)

⁢

\ar

⁢

@

⁢

3

-

[

r

]

-

⁢

F

11

⁢

&

⁢

(

c

21

,

c

12

,

…

,

c

1

⁢

n

)

⁢

\ar

⁢

@

⁢

3

-

[

r

]

-

⁢

F

12

⁢

&

⁢

⋯

⁢

\ar

⁢

@

⁢

3

-

[

r

]

-

⁢

F

1

⁢

n

⁢

&

⁢

(

c

21

,

c

22

,

…

,

c

2

⁢

n

)

The idea is to replace the

coordinates

one at a time, in sequence, from the first to the last, until all

n

coordinates are completely replaced from

(

c

11

,

…

,

c

1

⁢

n

)

to

(

c

21

,

…

,

c

2

⁢

n

)

.

Now, since each

F

1

⁢

j

is a congruence, apply

ω

to each

n

-tuple to get a new

\xymatrix

⁢

ω

⁢

(

c

11

,

c

12

,

…

,

c

1

⁢

n

)

⁢

\ar

⁢

@

⁢

3

-

[

r

]

-

⁢

F

11

⁢

&

⁢

ω

⁢

(

c

21

,

c

12

,

…

,

c

1

⁢

n

)

⁢

\ar

⁢

@

⁢

3

-

[

r

]

-

⁢

F

12

⁢

&

⁢

⋯

⁢

\ar

⁢

@

⁢

3

-

[

r

]

-

⁢

F

1

⁢

n

⁢

&

⁢

ω

⁢

(

c

21

,

c

22

,

…

,

c

2

⁢

n

)

.

But this chain implies that

ω

⁢

(

a

1

,

…

,

a

n

)

=

ω

⁢

(

c

11

,

…

,

c

i

⁢

n

)

≡

ω

⁢

(

c

21

,

…

,

c

2

⁢

n

)

(

mod

Θ

)

. So what we have shown is that the images of the first congruence pair are congruent modulo

Θ

. But this process can be easily applied to subsequent congruence pairs, so that we end up with

\xymatrix

⁢

ω

⁢

(

a

1

,

…

,

a

2

)

⁢

\ar

⁢

@

⁢

3

-

[

r

]

-

⁢

Θ

⁢

&

⁢

ω

⁢

(

c

21

,

…

,

c

2

⁢

n

)

⁢

\ar

⁢

@

⁢

3

-

[

r

]

-

⁢

Θ

⁢

&

⁢

⋯

⁢

\ar

⁢

@

⁢

3

-

[

r

]

-

⁢

Θ

⁢

&

⁢

ω

⁢

(

b

1

,

…

,

b

n

)

.

As

Θ

is an equivalence relation,

ω

⁢

(

a

1

,

…

,

a

2

)

≡

ω

⁢

(

b

1

,

…

,

b

n

)

(

mod

Θ

)

, completing the proof.
∎

Remarks

.

•

Furthermore, it can be shown that

Con

⁡

(

A

)

is an

algebraic lattice

. The

compact elements

of

Con

⁡

(

A

)

are finite joins of so-called

principal congruences

.

•

Conversely

, it can be shown (by Grätzer) that every algebraic lattice is

isomorphic

to the congruence lattice of some algebraic system.

•

If

A

is a lattice, then

Con

⁡

(

A

)

is

distributive

. The

converse

statement, that every distributive algebraic lattice is isomorphic to a congruence lattice, has recently been proven to be false by F. Wehrung.

References

1

G. Grätzer:

Universal Algebra

, 2nd Edition, Springer, New York (1978).

2

F. Wehrung:

http://hal.archives-ouvertes.fr/docs/00/11/93/14/PDF/CLP.pdf

A

Solution

to Dilworth’s Congruence Lattice Problem

, (2005).

Title

congruence lattice

Canonical name

CongruenceLattice

Date of creation

2013-03-22 17:06:31

Last modified on

2013-03-22 17:06:31

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

8

Author

CWoo (3771)

Entry type

Definition

Classification

msc 08A30

Defines

lattice of congruences

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