permutable congruences

permutable congruences

Let

A

be an

algebraic system

and

Θ

1

and

Θ

2

are two

congruences

on

A

.

Θ

1

and

Θ

2

are said to be

permutable

if

Θ

1

∘

Θ

2

=

Θ

2

∘

Θ

1

, where

∘

is the

composition of relations

.

For example, let

A

be the

direct product

of

A

1

and

A

2

. Define

Θ

1

on

A

as follows:

(

a

,

b

)

≡

(

c

,

d

)

(

mod

Θ

1

)

iff

a

=

c

.

Then

Θ

1

is clearly an

equivalence relation

on

A

. For any

n

-ary

operator

f

on

A

, let

f

1

and

f

2

be the corresponding

n

-ary operators on

A

1

and

A

2

respectively:

f

=

(

f

1

,

f

2

)

. Suppose

(

a

i

,

b

i

)

≡

(

c

i

,

d

i

)

(

mod

Θ

1

)

,

i

=

1

,

…

,

n

. Then

f

⁢

(

(

a

1

,

b

1

)

,

…

,

(

a

n

,

b

n

)

)

=

(

f

1

⁢

(

a

1

,

…

,

a

n

)

,

f

2

⁢

(

b

1

,

…

,

b

n

)

)

(1)

≡

(

f

1

⁢

(

c

1

,

…

,

c

n

)

,

f

2

⁢

(

d

1

,

…

,

d

n

)

)

(2)

=

f

⁢

(

(

c

1

,

d

1

)

,

…

,

(

c

n

,

d

n

)

)

(

mod

Θ

1

)

.

(3)

The equivalence of (1) and (2) follows from the

assumption

that

a

i

=

c

i

for each

i

=

1

,

…

,

n

, so that

f

1

⁢

(

a

1

,

…

,

a

n

)

=

f

1

⁢

(

c

1

,

…

,

c

n

)

. Similarly define

(

a

,

b

)

≡

(

c

,

d

)

(

mod

Θ

2

)

iff

b

=

d

.

By a

similar

argument

,

Θ

2

is a congruence on

A

too. Pick any

(

a

,

b

)

,

(

c

,

d

)

∈

A

. Then

(

a

,

b

)

≡

(

a

,

d

)

(

mod

Θ

1

)

and

(

a

,

d

)

≡

(

c

,

d

)

(

mod

Θ

2

)

so that

(

a

,

b

)

⁢

(

Θ

1

∘

Θ

2

)

⁢

(

c

,

d

)

. This implies that

Θ

1

∘

Θ

2

=

A

2

. Similarly

Θ

2

∘

Θ

1

=

A

2

. Therefore,

Θ

1

and

Θ

2

are permutable.

In fact, we have the following:

Proposition 1

.

Let

A

be an algebraic system with congruenes

Θ

1

and

Θ

2

. Then

Θ

1

and

Θ

2

are permutable iff

Θ

1

∘

Θ

2

=

Θ

1

∨

Θ

2

, where

∨

is the join

operation

on,

Con

⁡

(

A

)

, the

lattice of congruences

on

A

.

Proof.

Clearly, if

Θ

1

∘

Θ

2

=

Θ

1

∨

Θ

2

, then they are permutable.

Conversely

, suppose they are permutable. Let

C

=

Θ

1

∘

Θ

2

and

D

=

Θ

1

∨

Θ

2

. We want to show that

C

=

D

. If

(

a

,

b

)

∈

C

, then there is

c

∈

A

such that

(

a

,

c

)

∈

Θ

1

and

(

c

,

b

)

∈

Θ

2

, so

a

≡

b

(

mod

D

)

. This shows

C

⊆

D

. If

a

≡

b

(

mod

D

)

, then there is

c

∈

A

such that

a

≡

c

(

mod

R

)

and

c

≡

b

(

mod

S

)

with

R

,

S

∈

{

Θ

1

,

Θ

2

}

. If

R

=

S

, then we are done, since

Θ

i

⊆

C

(as an

element

(

a

,

b

)

belonging to, say

Θ

1

, can be written as

(

a

,

b

)

∘

(

b

,

b

)

∈

C

). If

R

=

Θ

1

and

S

=

Θ

2

then we are done too, since this is just the definition of

C

. If

R

=

Θ

2

and

S

=

Θ

1

, then

(

a

,

b

)

∈

Θ

2

∘

Θ

1

=

Θ

1

∘

Θ

2

=

C

, by permutability.
∎

Remark

. From the example above, it is not hard to see that an algebraic system

A

is the direct product of two algebraic systems

B

,

C

iff there are two

permutable congruences

Θ

and

Φ

on

A

such that

Θ

∨

Φ

=

A

2

and

Θ

∧

Φ

=

Δ

, where

Δ

=

{

(

a

,

a

)

∣

a

∈

A

}

is the

diagonal relation

on

A

, and that

B

≅

A

/

Θ

and

C

≅

A

/

Φ

. This result can be generalized to arbitrary direct products.

Title

permutable congruences

Canonical name

PermutableCongruences

Date of creation

2013-03-22 17:09:29

Last modified on

2013-03-22 17:09:29

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

6

Author

CWoo (3771)

Entry type

Definition

Classification

msc 08A30

Defines

completely permutable

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