congruence on a partial algebra

congruence on a partial algebra

Definition

There are two types of

congruences

on a

partial algebra

𝑨

, both are special types of a certain

equivalence relation

on

A

:

1.

Θ

is a

congruence relation

on

𝑨

if, given that

–

a

1

≡

b

1

(

mod

Θ

)

,

…

,

a

n

≡

b

n

(

mod

Θ

)

,

–

both

f

𝑨

⁢

(

a

1

,

…

,

a

n

)

and

f

𝑨

⁢

(

b

1

,

…

,

b

n

)

are defined,

then

f

𝑨

⁢

(

a

1

,

…

,

a

n

)

≡

f

𝑨

⁢

(

b

1

,

…

,

b

n

)

(

mod

Θ

)

.

2.

Θ

is a

strong congruence relation

on

𝑨

if it is a congruence relation on

𝑨

, and, given

–

a

1

≡

b

1

(

mod

Θ

)

,

…

,

a

n

≡

b

n

(

mod

Θ

)

,

–

f

𝑨

⁢

(

a

1

,

…

,

a

n

)

is defined,

then

f

𝑨

⁢

(

b

1

,

…

,

b

n

)

is defined.

Proposition 1

.

If

ϕ

:

𝐀

→

𝐁

is a

homomorphism

, then the equivalence relation

E

ϕ

induced by

ϕ

on

A

is a congruence relation. Furthermore, if

ϕ

is a strong, so is

E

ϕ

.

Proof.

Let

f

∈

τ

be an

n

-ary

function symbol

. Suppose

a

i

≡

b

i

(

mod

E

ϕ

)

and both

f

𝑨

⁢

(

a

1

,

…

,

a

n

)

and

f

𝑨

⁢

(

b

1

,

…

,

b

n

)

are defined. Then

ϕ

⁢

(

a

i

)

=

ϕ

⁢

(

b

i

)

, and therefore

ϕ

⁢

(

f

𝑨

⁢

(

a

1

,

…

,

a

n

)

)

=

f

𝑩

⁢

(

ϕ

⁢

(

a

1

)

,

…

,

ϕ

⁢

(

a

n

)

)

=

f

𝑩

⁢

(

ϕ

⁢

(

b

1

)

,

…

,

ϕ

⁢

(

b

n

)

)

=

ϕ

⁢

(

f

𝑨

⁢

(

b

1

,

…

,

b

n

)

)

,

so

f

𝑨

⁢

(

a

1

,

…

,

a

n

)

≡

f

𝑨

⁢

(

b

1

,

…

,

b

n

)

(

mod

E

ϕ

)

. In other words,

E

ϕ

is a congruence relation.

Now, suppose in

addition

that

ϕ

is a strong homomorphism. Again, let

a

i

≡

b

i

(

mod

E

ϕ

)

. Assume

f

𝑨

⁢

(

a

1

,

…

,

a

n

)

is defined. Since

ϕ

⁢

(

a

i

)

=

ϕ

⁢

(

b

i

)

, we get

ϕ

⁢

(

f

𝑨

⁢

(

a

1

,

…

,

a

n

)

)

=

f

𝑩

⁢

(

ϕ

⁢

(

a

1

)

,

…

,

ϕ

⁢

(

a

n

)

)

=

f

𝑩

⁢

(

ϕ

⁢

(

b

1

)

,

…

,

ϕ

⁢

(

b

n

)

)

.

Since

ϕ

is strong,

f

𝑨

⁢

(

b

1

,

…

,

b

n

)

is defined, which means that

E

ϕ

is strong.
∎

Congruences as Subalgebras

If

𝑨

is a partial algebra of type

τ

, then the

direct power

𝑨

2

is a partial algebra of type

τ

. A

binary relation

Θ

on

A

may be viewed as a subset of

A

2

. For each

n

-ary

operation

f

𝑨

2

on

𝑨

2

, take the

restriction

on

Θ

, and call it

f

𝚯

. For

a

i

∈

Θ

,

f

𝚯

⁢

(

a

1

,

…

,

a

n

)

is defined in

Θ

iff

f

𝑨

2

⁢

(

a

1

,

…

,

a

n

)

is defined at all, and its value is in

Θ

. When

f

𝚯

⁢

(

a

1

,

…

,

a

n

)

is defined in

Θ

, its value is set as

f

𝑨

2

⁢

(

a

1

,

…

,

a

n

)

. This turns

𝚯

into a partial algebra. However, the type of

𝚯

is

τ

only when

f

𝚯

is non-empty for each function symbol

f

∈

τ

. In particular,

Proposition 2

.

If

Θ

is

reflexive

, then

Θ

is a

relative subalgebra

of

𝐀

2

.

Proof.

Pick any

n

-ary function symbol

f

∈

τ

. Then

f

𝑨

⁢

(

a

1

,

…

,

a

n

)

is defined for some

a

i

∈

A

. Then

f

𝑨

2

⁢

(

(

a

1

,

a

1

)

,

…

,

(

a

n

,

a

n

)

)

is defined and is equal to

(

f

𝑨

⁢

(

a

1

,

…

,

a

n

)

,

f

𝑨

⁢

(

a

1

,

…

,

a

n

)

)

, which is in

Θ

, since

Θ

is reflexive. This shows that

f

𝚯

⁢

(

(

a

1

,

a

1

)

,

…

,

(

a

n

,

a

n

)

)

is defined. As a result,

𝚯

is a partial algebra of type

τ

. Furthermore, by virtue of the way

f

𝚯

is defined for each

f

∈

τ

,

𝚯

is a relative subalgebra of

𝑨

.
∎

Proposition 3

.

An equivalence relation

Θ

on

A

is a congruence iff

Θ

is a

subalgebra

of

𝐀

2

.

Proof.

First, assume that

Θ

is a congruence relation on

A

. Since

Θ

is reflexive,

𝚯

is a relative subalgebra of

𝑨

2

. Now, suppose

f

𝑨

2

⁢

(

(

a

1

,

b

1

)

,

…

,

(

a

n

,

b

n

)

)

exists, where

a

i

≡

b

i

(

mod

Θ

)

. Then

f

𝑨

⁢

(

a

1

,

…

,

a

n

)

,

f

𝑨

⁢

(

b

1

,

…

,

b

n

)

both exist. Since

Θ

is a congruence,

f

𝑨

⁢

(

a

1

,

…

,

a

n

)

≡

f

𝑨

⁢

(

b

1

,

…

,

b

n

)

(

mod

Θ

)

. In other words,

(

f

𝑨

⁢

(

a

1

,

…

,

a

n

)

,

f

𝑨

⁢

(

b

1

,

…

,

b

n

)

)

∈

Θ

. Hence

𝚯

is a subalgebra of

𝑨

2

.

Conversely

, assume

𝚯

is a subalgebra of

𝑨

2

. Suppose

(

a

i

,

b

i

)

∈

Θ

and both

f

𝑨

⁢

(

a

1

,

…

,

a

n

)

and

f

𝑨

⁢

(

b

1

,

…

,

b

n

)

are defined. Then

f

𝑨

2

⁢

(

(

a

1

,

b

1

)

,

…

,

(

a

n

,

b

n

)

)

is defined. Since

𝚯

is a subalgebra of

𝑨

2

,

f

𝚯

⁢

(

(

a

1

,

b

1

)

,

…

,

(

a

n

,

b

n

)

)

is also defined, and

(

f

𝑨

⁢

(

a

1

,

…

,

a

n

)

,

f

𝑨

⁢

(

b

1

,

…

,

b

n

)

)

=

f

𝑨

2

⁢

(

(

a

1

,

b

1

)

,

…

,

(

a

n

,

b

n

)

)

=

f

𝚯

⁢

(

(

a

1

,

b

1

)

,

…

,

(

a

n

,

b

n

)

)

∈

Θ

. This shows that

Θ

is a congruence relation on

A

.
∎

Quotient Partial Algebras

With congruence relations defined, one may then define quotient partial algebras: given a partial algebra

𝑨

of type

τ

and a congruence relation

Θ

on

A

, the

quotient partial algebra

of

𝑨

by

Θ

is the partial algebra

𝑨

/

𝚯

whose underlying set is

A

/

Θ

, the set of congruence classes, and for each

n

-ary function symbol

f

∈

τ

,

f

𝑨

/

𝚯

⁢

(

[

a

1

]

,

…

,

[

a

n

]

)

is defined iff there are

b

1

,

…

,

b

n

∈

A

such that

[

a

i

]

=

[

b

i

]

and

f

𝑨

⁢

(

b

1

,

…

,

b

n

)

is defined. When this is the case:

f

𝑨

/

𝚯

⁢

(

[

a

1

]

,

…

,

[

a

n

]

)

:=

[

f

𝑨

⁢

(

b

1

,

…

,

b

n

)

]

.

Suppose there are

c

1

,

…

,

c

n

∈

A

such that

[

a

i

]

=

[

c

i

]

, or

a

i

≡

c

i

(

mod

Θ

)

, and

f

𝑨

⁢

(

c

1

,

…

,

c

n

)

is defined, then

b

i

≡

c

i

(

mod

Θ

)

and

f

𝑨

⁢

(

b

1

,

…

,

b

n

)

≡

f

𝑨

⁢

(

c

1

,

…

,

c

n

)

(

mod

Θ

)

, or, equivalently,

[

f

𝑨

⁢

(

b

1

,

…

,

b

n

)

]

=

[

f

𝑨

⁢

(

c

1

,

…

,

c

n

)

]

, so that

f

𝑨

/

𝚯

is a well-defined operation.

In addition, it is easy to see that

𝑨

/

𝚯

is in fact a

τ

-

algebra

. For each

n

-ary

f

∈

τ

, pick

a

1

,

…

,

a

n

∈

A

such that

f

𝑨

⁢

(

a

1

,

…

,

a

n

)

is defined. Then

f

𝑨

/

𝚯

⁢

(

[

a

1

]

,

…

,

[

a

n

]

)

is defined, and is equal to

[

f

𝑨

⁢

(

a

1

,

…

,

a

n

)

]

.

Proposition 4

.

Let

𝐀

and

Θ

be defined as above. Then

[

⋅

]

:

𝐀

→

𝐀

/

Θ

, given by

[

⋅

]

⁢

(

a

)

=

[

a

]

, is a

surjective

full homomorphism, and

E

[

⋅

]

=

Θ

. Furthermore,

[

⋅

]

is a strong homomorphism iff

Θ

is a strong congruence relation.

Proof.

[

⋅

]

is obviously surjective. The fact that

[

⋅

]

is a full homomorphism follows directly from the definition of

f

𝑨

/

𝚯

, for each

f

∈

τ

. Next,

a

⁢

E

[

⋅

]

⁢

b

iff

[

a

]

=

[

b

]

iff

a

≡

b

(

mod

Θ

)

. This proves the first statement.

The next statement is proved as follows:

(

⇒

)

. If

a

i

≡

b

i

(

mod

Θ

)

and

f

𝑨

⁢

(

a

1

,

…

,

a

n

)

is defined, then

f

𝑨

/

𝚯

⁢

(

[

a

1

]

,

…

,

[

a

n

]

)

is defined, which is just

f

𝑨

/

𝚯

⁢

(

[

b

1

]

,

…

,

[

b

n

]

)

, and, as

[

⋅

]

is strong,

f

𝑨

⁢

(

b

1

,

…

,

b

n

)

is defined, showing that

Θ

is strong.

(

⇐

)

. Suppose

f

𝑨

/

𝚯

⁢

(

[

a

1

]

,

…

,

[

a

n

]

)

is defined. Then there are

b

1

,

…

,

b

n

∈

A

with

a

i

≡

b

i

(

mod

Θ

)

such that

f

𝑨

⁢

(

b

1

,

…

,

b

n

)

is defined. Since

Θ

is strong,

f

𝑨

⁢

(

a

1

,

…

,

a

n

)

is defined as well, which shows that

[

⋅

]

is strong.
∎

References

1

G. Grätzer:

Universal Algebra

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

Title

congruence on a partial algebra

Canonical name

CongruenceOnAPartialAlgebra

Date of creation

2013-03-22 18:43:01

Last modified on

2013-03-22 18:43:01

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

14

Author

CWoo (3771)

Entry type

Definition

Classification

msc 08A55

Classification

msc 03E99

Classification

msc 08A62

Synonym

congruence

Synonym

strong congruence

Synonym

quotient algebra

Defines

congruence relation

Defines

strong congruence relation

Defines

quotient partial algebra

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