reduced direct product

reduced direct product

Let

{

A

i

∣

i

∈

I

}

be a set of

algebraic systems

of the same type,

indexed by

I

. Let

A

be the

direct product

of the

A

i

’s. For any

a

,

b

∈

A

, set

supp

⁡

(

a

,

b

)

:=

{

k

∈

I

∣

a

⁢

(

k

)

≠

b

⁢

(

k

)

}

.

Consider a

Boolean ideal

L

of the

Boolean algebra

P

⁢

(

I

)

of

I

. Define a

binary relation

Θ

L

on

A

as follows:

(

a

,

b

)

∈

Θ

L

iff

supp

⁡

(

a

,

b

)

∈

L

.

Lemma 1

.

Θ

L

defined above is a

congruence relation

on

A

.

Proof.

Since

L

is an ideal

∅

∈

L

. Therefore,

(

a

,

a

)

∈

Θ

L

, since

{

k

∈

I

∣

a

⁢

(

k

)

≠

a

⁢

(

k

)

}

=

∅

. Clearly,

Θ

L

is

symmetric

. For

transitivity

, suppose

(

a

,

b

,

(

b

,

c

)

∈

Θ

L

. If

a

⁢

(

k

)

≠

c

⁢

(

k

)

for some

k

∈

I

, then either

a

⁢

(

k

)

≠

b

⁢

(

k

)

or

b

⁢

(

k

)

≠

c

⁢

(

k

)

(a

contrapositive

argument

). So

supp

⁡

(

a

,

c

)

⊆

supp

⁡

(

a

,

b

)

∪

supp

⁡

(

b

,

c

)

.

Since

L

is an ideal,

supp

⁡

(

a

,

c

)

∈

L

, so

(

a

,

c

)

∈

Θ

L

, and

Θ

L

is an

equivalence relation

on

A

.

Next, let

ω

be an

n

-ary

operator

on

A

and

a

j

≡

b

j

(

mod

Θ

L

)

, where

j

=

1

,

…

,

n

. We want to show that

ω

⁢

(

a

1

,

…

,

a

n

)

≡

ω

⁢

(

b

1

,

…

,

b

n

)

(

mod

Θ

L

)

. Let

ω

i

be the associated

n

-ary operators on

A

i

. If

ω

⁢

(

a

1

,

…

,

a

n

)

⁢

(

k

)

≠

ω

⁢

(

b

1

,

…

,

b

n

)

⁢

(

k

)

, then

ω

k

⁢

(

a

1

⁢

(

k

)

,

…

,

a

n

⁢

(

k

)

)

≠

ω

k

⁢

(

b

1

⁢

(

k

)

,

…

,

b

n

⁢

(

k

)

)

, which implies that

a

j

⁢

(

k

)

≠

b

j

⁢

(

k

)

for some

j

=

1

,

…

,

n

. This implies that

supp

⁡

(

ω

⁢

(

a

1

,

…

,

a

n

)

,

ω

⁢

(

b

1

,

…

,

b

n

)

)

⊆

⋃

j

=

1

n

supp

⁡

(

a

j

,

b

j

)

.

Since

L

is an ideal, and each

supp

⁡

(

a

j

,

b

j

)

∈

L

, we have that

supp

⁡

(

ω

⁢

(

a

1

,

…

,

a

n

)

,

ω

⁢

(

b

1

,

…

,

b

n

)

)

∈

L

as well, this means that

ω

⁢

(

a

1

,

…

,

a

n

)

≡

ω

⁢

(

b

1

,

…

,

b

n

)

(

mod

Θ

L

)

.
∎

Definition

. Let

A

=

∏

{

A

i

∣

i

∈

I

}

,

L

be a Boolean ideal of

P

⁢

(

I

)

and

Θ

L

be defined as above. The quotient algebra

A

/

Θ

L

is called the

L

-

reduced direct product

of

A

i

. The

L

-reduced direct product of

A

i

is denoted by

∏

L

{

A

i

∣

i

∈

I

}

. Given any

element

a

∈

A

, its image in the reduced direct product

∏

L

{

A

i

∣

i

∈

I

}

is given by

[

a

]

⁢

Θ

L

, or

[

a

]

for short.

Example

. Let

A

=

A

1

×

⋯

×

A

n

, and let

L

be the

principal ideal

generated by

1

. Then

L

=

{

∅

,

{

1

}

}

. The

congruence

Θ

L

is given by

(

a

1

,

…

,

a

n

)

≡

(

b

1

,

…

,

b

n

)

(

mod

Θ

L

)

iff

{

i

∣

a

i

≠

b

i

}

=

∅

or

{

1

}

. This implies that

a

i

=

b

i

for all

i

=

2

,

…

,

n

. In other words,

Θ

L

is

isomorphic

to the direct product of

A

2

×

⋯

×

A

n

. Therefore, the

L

-reduced direct product of

A

i

is isomorphic to

A

1

.

The example above can be generalized: if

J

⊆

I

, then

∏

{

A

i

∣

i

∈

I

}

P

⁢

(

J

)

≅

∏

{

A

i

∣

i

∈

I

-

J

}

.

For

a

∈

A

=

∏

{

A

i

∣

i

∈

I

}

, write

a

=

(

a

i

)

i

∈

I

. It is not hard to see that the map

f

:

∏

P

⁢

(

J

)

{

A

i

∣

i

∈

I

}

→

∏

{

A

i

∣

i

∈

I

-

J

}

given by

f

⁢

(

[

a

]

)

=

(

a

i

)

i

∈

I

-

J

is the required

isomorphism

.

Remark

. The definition of a reduced direct product in terms of a Boolean ideal can be equivalently stated in terms of a Boolean filter

F

. All there is to do is to replace

supp

⁡

(

a

,

b

)

by its

complement

:

supp

(

a

,

b

)

c

:=

{

k

∈

I

∣

a

(

k

)

=

b

(

k

)

}

. The congruence relation is now

Θ

F

′

, where

F

′

=

{

I

-

J

∣

J

∈

F

}

is the ideal complement of

F

. When

F

is prime, the

F

′

-reduced direct product is called a

prime product

, or an

ultraproduct

, since any

prime filter

is also called an

ultrafilter

. Ultraproducts can be more generally defined over arbitrary

structures

.

References

1

G. Grätzer:

Universal Algebra

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

Title

reduced direct product

Canonical name

ReducedDirectProduct

Date of creation

2013-03-22 17:10:11

Last modified on

2013-03-22 17:10:11

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

10

Author

CWoo (3771)

Entry type

Definition

Classification

msc 08B25

Synonym

ultraproduct

Defines

prime product

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