direct limit of algebraic systems

direct limit of algebraic systems

An immediate

generalization

of the

concept

of the

direct limit

of a

direct family

of sets is the direct limit of a

direct family of algebraic systems

.

Direct Family of Algebraic Systems

The definition is almost identical to that of a direct family of sets, except that

functions

ϕ

i

⁢

j

are now

homomorphisms

. For completeness, we will spell out the definition in its entirety.

Let

𝒜

=

{

A

i

∣

i

∈

I

}

be a family of

algebraic systems

of the same type (say, they are all

O

-

algebras

),

indexed by

a non-empty set

I

.

𝒜

is said to be a

direct family

if

1.

I

is a

directed set

,

2.

whenever

i

≤

j

in

I

, there is a homomorphism

ϕ

i

⁢

j

:

A

i

→

A

j

,

3.

ϕ

i

⁢

i

is the

identity

on

A

i

,

4.

if

i

≤

j

≤

k

, then

ϕ

j

⁢

k

∘

ϕ

i

⁢

j

=

ϕ

i

⁢

k

.

An example of this is a direct family of sets. A homomorphism between two sets is just a function between the sets.

Direct Limit of Algebraic Systems

Let

𝒜

be a direct family of algebraic systems

A

i

, indexed by

I

(

i

∈

I

). Take the

disjoint union

of the underlying sets of each algebraic system, and call it

A

. Next, a

binary relation

∼

is defined on

A

as
follows:

given that

a

∈

A

i

and

b

∈

A

j

,

a

∼

b

iff there is

A

k

such that

ϕ

i

⁢

k

⁢

(

a

)

=

ϕ

j

⁢

k

⁢

(

b

)

.

It is shown here (

http://planetmath.org/DirectLimitOfSets

) that

∼

is an

equivalence relation

on

A

, so we can take the

quotient

A

/

∼

, and denote it by

A

∞

. Elements of

A

∞

are denoted by

[

a

]

I

or

[

a

]

when there is no confusion, where

a

∈

A

. So

A

∞

is just the direct limit of

A

i

considered as sets

.

Next, we want to turn

A

∞

into an

O

-algebra. Corresponding to each set of

n

-ary

operations

ω

i

defined on

A

i

for all

i

∈

I

, we define an

n

-ary operation

ω

on

A

∞

as follows:

for

i

=

1

,

…

,

n

, pick

a

i

∈

A

j

⁢

(

i

)

,

j

⁢

(

i

)

∈

I

. Let

J

:=

{

j

⁢

(

i

)

∣

i

=

1

,

…

,

n

}

. Since

I

is directed and

J

is finite,

J

has an

upper bound

j

∈

I

. Let

α

i

=

ϕ

j

⁢

(

i

)

⁢

j

⁢

(

a

i

)

. Define

ω

⁢

(

[

a

1

]

,

…

,

[

a

n

]

)

:=

[

ω

j

⁢

(

α

1

,

…

,

α

n

)

]

.

Proposition 1

.

ω

is a well-defined

n

-ary operation on

A

∞

.

Proof.

Suppose

[

b

1

]

=

[

a

1

]

,

…

,

[

b

n

]

=

[

a

n

]

. Let

α

i

be defined as above, and let

a

:=

ω

j

⁢

(

α

1

,

…

,

α

n

)

∈

A

j

. Similarly,

β

i

are defined:

β

i

:=

ϕ

k

⁢

(

i

)

⁢

k

⁢

(

b

i

)

∈

A

k

, where

b

i

∈

A

k

⁢

(

i

)

. Let

b

:=

ω

k

⁢

(

β

1

,

…

,

β

n

)

∈

A

k

. We want to show that

a

∼

b

.

Since

a

i

∼

b

i

,

α

i

∼

β

i

. So there is

c

i

:=

ϕ

j

⁢

ℓ

⁢

(

i

)

⁢

(

α

i

)

=

ϕ

k

⁢

ℓ

⁢

(

i

)

⁢

(

β

i

)

∈

A

ℓ

⁢

(

i

)

. Let

ℓ

be the upper bound of the set

{

ℓ

⁢

(

1

)

,

…

,

ℓ

⁢

(

n

)

}

and define

γ

i

:=

ϕ

ℓ

⁢

(

i

)

⁢

ℓ

⁢

(

c

i

)

∈

A

ℓ

. Then

ϕ

j

⁢

ℓ

⁢

(

a

)

=

ϕ

j

⁢

ℓ

⁢

(

ω

j

⁢

(

α

1

,

…

,

α

n

)

)

=

ω

ℓ

⁢

(

ϕ

j

⁢

ℓ

⁢

(

α

1

)

,

…

,

ϕ

j

⁢

ℓ

⁢

(

α

n

)

)

=

ω

ℓ

⁢

(

ϕ

ℓ

⁢

(

1

)

⁢

ℓ

∘

ϕ

j

⁢

ℓ

⁢

(

1

)

⁢

(

α

1

)

,

…

,

ϕ

ℓ

⁢

(

n

)

⁢

ℓ

∘

ϕ

j

⁢

ℓ

⁢

(

n

)

⁢

(

α

n

)

)

=

ω

ℓ

⁢

(

ϕ

ℓ

⁢

(

1

)

⁢

ℓ

⁢

(

c

1

)

,

…

,

ϕ

ℓ

⁢

(

n

)

⁢

ℓ

⁢

(

c

n

)

)

=

ω

ℓ

⁢

(

ϕ

ℓ

⁢

(

1

)

⁢

ℓ

∘

ϕ

k

⁢

ℓ

⁢

(

1

)

⁢

(

β

1

)

,

…

,

ϕ

ℓ

⁢

(

n

)

⁢

ℓ

∘

ϕ

k

⁢

ℓ

⁢

(

n

)

⁢

(

β

n

)

)

=

ω

ℓ

⁢

(

ϕ

k

⁢

ℓ

⁢

(

β

1

)

,

…

,

ϕ

k

⁢

ℓ

⁢

(

β

n

)

)

=

ϕ

k

⁢

ℓ

⁢

(

ω

k

⁢

(

β

1

,

…

,

β

n

)

)

=

ϕ

k

⁢

ℓ

⁢

(

b

)

,

which shows that

a

∼

b

.
∎

Definition

. Let

𝒜

be a direct family of algebraic systems of the same type (say

O

) indexed by

I

. The

O

-algebra

A

∞

constructed above is called the

direct limit

of

𝒜

.

A

∞

is alternatively written

lim

→

⁡

A

i

.

Remark

. Dually, one can define an

inverse family of algebraic systems

, and its

inverse limit

. The inverse limit of an inverse family

𝒜

is written

A

∞

or

lim

←

⁡

A

i

.

Title

direct limit of algebraic systems

Canonical name

DirectLimitOfAlgebraicSystems

Date of creation

2013-03-22 16:53:56

Last modified on

2013-03-22 16:53:56

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

7

Author

CWoo (3771)

Entry type

Definition

Classification

msc 08B25

Synonym

direct system of algebraic systems

Synonym

inverse system of algebraic systems

Synonym

projective system of algebraic systems

Related topic

DirectLimitOfSets

Defines

direct family of algebraic systems

Defines

inverse family of algebraic systems

Defines

inverse limit of algebraic systems

Generated on Fri Feb 9 14:05:54 2018 by

LaTeXML