congruence relation on an algebraic system

congruence relation on an algebraic system

Let

(

A

,

O

)

be an

algebraic system

. A

congruence relation

, or simply a

congruence

ℭ

on

A

1.

is an

equivalence relation

on

A

; if

(

a

,

b

)

∈

ℭ

we write

a

≡

b

(

mod

ℭ

)

, and

2.

respects every

n

-ary

operator

on

A

: if

ω

A

is an

n

-ary operator on

A

(

ω

∈

O

), and for any

a

i

,

b

i

∈

A

,

i

=

1

,

…

,

n

, we have

a

i

≡

b

i

(

mod

ℭ

)

implies

ω

A

⁢

(

a

1

,

…

,

a

n

)

≡

ω

A

⁢

(

b

1

,

…

,

b

n

)

(

mod

ℭ

)

.

For example,

A

2

and

Δ

A

:=

{

(

a

,

a

)

∣

a

∈

A

}

are both congruence relations on

A

.

Δ

A

is called the

trivial congruence

(on

A

). A

proper

congruence relation is one not equal to

A

2

.

Remarks

.

•

ℭ

is a congruence relation on

A

if and only if

ℭ

is an equivalence relation on

A

and a

subalgebra

of the

product

(

http://planetmath.org/DirectProductOfAlgebras

)

A

×

A

.

•

The set of congruences of an algebraic system is a

complete lattice

. The meet is the usual set

intersection

. The join (of an arbitrary number of congruences) is the join of the underlying equivalence relations (

http://planetmath.org/PartitionsFormALattice

). This join corresponds to the subalgebra (of

A

×

A

) generated by the union of the underlying sets of the congruences. The

lattice of congruences

on

A

is denoted by

Con

⁡

(

A

)

.

•

(

restriction

)

If

ℭ

is a congruence on

A

and

B

is a subalgebra of

A

, then

ℭ

B

defined by

ℭ

∩

(

B

×

B

)

is a congruence on

B

. The equivalence of

ℭ

B

is obvious. For any

n

-ary operator

ω

B

inherited from

A

’s

ω

A

, if

a

i

≡

b

i

(

mod

ℭ

B

)

, then

ω

B

⁢

(

a

1

,

…

,

a

n

)

=

ω

A

⁢

(

a

1

,

…

,

a

n

)

≡

ω

A

⁢

(

b

1

,

…

,

b

n

)

=

ω

B

⁢

(

b

1

,

…

,

b

n

)

(

mod

ℭ

)

. Since both

ω

B

⁢

(

a

1

,

…

,

a

n

)

and

ω

B

⁢

(

b

1

,

…

,

b

n

)

are in

B

,

ω

B

⁢

(

a

1

,

…

,

a

n

)

≡

ω

B

⁢

(

b

1

,

…

,

b

n

)

(

mod

ℭ

B

)

as well.

ℭ

B

is the congruence

restricted

to

B

.

•

(

extension

)

Again, let

ℭ

be a congruence on

A

and

B

a subalgebra of

A

. Define

B

ℭ

by

{

a

∈

A

∣

(

a

,

b

)

∈

ℭ

⁢

and

⁢

b

∈

B

}

. In other words,

a

∈

B

ℭ

iff

a

≡

b

(

mod

ℭ

)

for some

b

∈

B

. We assert that

B

ℭ

is a subalgebra of

A

. If

ω

A

is an

n

-ary operator on

A

and

a

1

,

…

,

a

n

∈

B

ℭ

, then

a

i

≡

b

i

(

mod

ℭ

)

, so

ω

A

⁢

(

a

1

,

…

,

a

n

)

≡

ω

A

⁢

(

b

1

,

…

,

b

n

)

(

mod

ℭ

)

. Since

ω

A

⁢

(

b

1

,

…

,

b

n

)

∈

B

,

ω

A

⁢

(

a

1

,

…

,

a

n

)

∈

B

ℭ

. Therefore,

B

ℭ

is a subalgebra. Because

B

⊆

B

ℭ

, we call it the

extension

of

B

by

ℭ

.

•

Let

B

be a subset of

A

×

A

. The smallest congruence

ℭ

on

A

such that

a

≡

b

(

mod

ℭ

)

for all

a

,

b

∈

B

is called the

congruence generated by

B

.

ℭ

is often written

⟨

B

⟩

. When

B

is a singleton

{

(

a

,

b

)

}

, then we call

⟨

B

⟩

a

principal congruence

, and denote it by

⟨

(

a

,

b

)

⟩

.

Quotient algebra

Given an

algebraic structure

(

A

,

O

)

and a congruence relation

ℭ

on

A

, we can

construct

a new

O

-

algebra

(

A

/

ℭ

,

O

)

, as follows: elements of

A

/

ℭ

are of the form

[

a

]

, where

a

∈

A

. We set

[

a

]

=

[

b

]

⁢

iff

⁢

a

≡

b

(

mod

ℭ

)

.

Furthermore, for each

n

-ary operator

ω

A

on

A

, define

ω

A

/

ℭ

by

ω

A

/

ℭ

⁢

(

[

a

1

]

,

…

,

[

a

n

]

)

:=

[

ω

A

⁢

(

a

1

,

…

,

a

n

)

]

.

It is easy to see that

ω

A

/

ℭ

is a well-defined operator on

A

/

ℭ

. The

O

-algebra thus constructed is called the

quotient algebra

of

A

over

ℭ

.

Remark

. The bracket

[

⋅

]

:

A

→

A

/

ℭ

is in fact an

epimorphism

, with kernel (

http://planetmath.org/KernelOfAHomomorphismBetweenAlgebraicSystems

)

ker

⁡

(

[

⋅

]

)

=

ℭ

. This means that every congruence of an algebraic system

A

is the kernel of some

homomorphism

from

A

.

[

⋅

]

is usually written

[

⋅

]

ℭ

to signify its association with

ℭ

.

References

1

G. Grätzer:

Universal Algebra

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

Title

congruence relation on an algebraic system

Canonical name

CongruenceRelationOnAnAlgebraicSystem

Date of creation

2013-03-22 16:26:23

Last modified on

2013-03-22 16:26:23

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

33

Author

CWoo (3771)

Entry type

Definition

Classification

msc 08A30

Related topic

Congruence3

Related topic

Congruence2

Related topic

CongruenceInAlgebraicNumberField

Related topic

PolynomialCongruence

Related topic

QuotientCategory

Related topic

CategoryOfAdditiveFractions

Defines

congruence

Defines

congruence relation

Defines

quotient algebra

Defines

proper congruence

Defines

trivial congruence

Defines

non-trivial congruence

Defines

congruence restricted to a subalgebra

Defines

extension of a subalgebra by a congruence

Defines

principal congruence

Defines

congruence generated by

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