aarjaneiro/planetmath huggingface dataset:
linear continuum

linear continuum

Let

X

be a

totally-ordered set

under an order

<

having at least two distinct points. Then

X

is said to be a

linear continuum

if the following two conditions are satisfied:

1.

The order relation

<

is a

dense total order

(i.e., for every

x

,

y

∈

X

with

x

<

y

there exists

z

∈

X

such that

x

<

z

<

y

).

2.

Every non-empty subset of

X

that is bounded above has a least

upper bound

(i.e.,

X

has the least upper bound property).

Some examples of

ordered sets

that are linear continua include

ℝ

, the set

[

0

,

1

]

×

[

0

,

1

]

in the

dictionary order

, and the so-called

long line

Ω

×

[

0

,

1

)

in the dictionary

topology

. (The third example is a special case of a general result on

well-ordered sets

and linear continua.)

Proposition

.

If

X

is a well-ordered set, then the set

X

×

[

0

,

1

)

is a linear continua in the dictionary order topology.

Linear continua are of special interest when they are made into

topological spaces

under the

order topology

, and the following two establish some useful properties of such spaces:

Proposition

.

If

X

is a linear continuum in the order topology, then

X

is

http://planetmath.org/node/941

connected

and so are

intervals

in

X

.

As a corollary of the preceding , we obtain the result that

ℝ

is in its

usual topology

, as are the intervals

[

a

,

b

]

and

(

a

,

b

)

, where

a

<

b

∈

ℝ

.

Proposition

.

If

X

is a linear continuum in the order topology, then every closed interval in

X

is

compact

.

Proof.

This is essentially a slightly generalized version of the

Heine-Borel Theorem

for

ℝ

, and the proof is almost identical.
∎

Title

linear continuum

Canonical name

LinearContinuum

Date of creation

2013-03-22 17:17:40

Last modified on

2013-03-22 17:17:40

Owner

azdbacks4234 (14155)

Last modified by

azdbacks4234 (14155)

Numerical id

11

Author

azdbacks4234 (14155)

Entry type

Definition

Classification

msc 06F30

Classification

msc 54B99

Related topic

DenseTotalOrder

Related topic

TotalOrder

Related topic

Supremum

Related topic

LowestUpperBound

Related topic

OrderTopology

Related topic

ASpaceIsConnectedUnderTheOrderedTopologyIfAndOnlyIfItIsALinearContinuum

Defines

linear continuum

Generated on Fri Feb 9 10:06:54 2018 by

LaTeXML
latticoid

latticoid

A

latticoid

is a set

L

with two

binary operations

, the meet

∧

and the join

∨

on

L

satisfying the following conditions:

1.

(idempotence)

x

∧

x

=

x

∨

x

=

x

for any

x

∈

L

,

2.

(

commutativity

)

x

∧

y

=

y

∧

x

and

x

∨

y

=

y

∨

x

for any

x

,

y

∈

L

, and

3.

(

absorption

)

x

∨

(

y

∧

x

)

=

x

∧

(

y

∨

x

)

=

x

for any

x

,

y

∈

L

.

A latticoid is like a

lattice

without the

associativity

assumption

, i.e., a lattice is a latticoid that is both meet associative and join associative.

If one of the binary operations is associative, say

∧

is associative, we may define a latticoid as a poset as follows:

x

≤

y

⁢

iff

⁢

x

∧

y

=

x

.

Clearly,

≤

is

reflexive

, as

x

∧

x

=

x

. If

x

≤

y

and

y

≤

x

, then

x

=

x

∧

y

=

y

∧

x

=

y

, so

≤

is anti-symmetric. Finally, suppose

x

≤

y

and

y

≤

z

, then

x

∧

z

=

(

x

∧

y

)

∧

z

=

x

∧

(

y

∧

z

)

=

x

∧

y

=

x

, or

x

≤

z

,

≤

is

transitive

.

Once a latticoid is a poset, we may easily visualize it by a diagram (

Hasse diagram

), much like that of a lattice. Position

y

above

x

if

x

≤

y

and connect a

line segment

between

x

and

y

. The following is the diagram of a latticoid that is meet associative but not join associative:

\xymatrix

⁢

&

⁢

a

∨

b

⁢

\ar

⁢

@

-

[

d

]

⁢

&

⁢

&

⁢

c

⁢

\ar

⁢

@

-

[

l

⁢

d

]

⁢

\ar

⁢

@

-

[

r

⁢

d

]

⁢

&

⁢

a

⁢

\ar

⁢

@

-

[

r

⁢

d

]

⁢

&

⁢

&

⁢

b

⁢

\ar

⁢

@

-

[

l

⁢

d

]

⁢

&

⁢

a

∧

b

⁢

&

It is not join associative because

(

a

∨

b

)

∨

c

=

a

∨

b

, whereas

a

∨

(

b

∨

c

)

=

a

∨

c

=

c

≠

a

∨

b

.

Given a latticoid

L

, we can define a dual

L

*

of

L

by using the same underlying set, and define the meet of

a

and

b

in

L

*

as the join of

a

and

b

in

L

, and the join of

a

and

b

(in

L

*

) as the meet of

a

and

b

in

L

.

L

is a meet-associative latticoid iff

L

*

is join-associative.

Title

latticoid

Canonical name

Latticoid

Date of creation

2013-03-22 16:31:02

Last modified on

2013-03-22 16:31:02

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

8

Author

CWoo (3771)

Entry type

Definition

Classification

msc 06F99

Generated on Fri Feb 9 10:07:06 2018 by

LaTeXML
direct product of algebras

direct product of algebras

In this entry, let

O

be a fixed

operator set

. All algebraic systems have the same type (they are all

O

-

algebras

).

Let

{

A

i

∣

i

∈

I

}

be a set of algebraic systems of the same type (

O

)

indexed by

I

. Let us form the

Cartesian product

of the underlying sets and call it

A

:

A

:=

∏

i

∈

I

A

i

.

Recall that

element

a

of

A

is a

function

from

I

to

⋃

A

i

such that for each

i

∈

I

,

a

⁢

(

i

)

∈

A

i

.

For each

ω

∈

O

with arity

n

, let

ω

A

i

be the corresponding

n

-ary

operator

on

A

i

. Define

ω

A

:

A

n

→

A

by

ω

A

⁢

(

a

1

,

…

,

a

n

)

⁢

(

i

)

=

ω

A

i

⁢

(

a

1

⁢

(

i

)

,

…

,

a

n

⁢

(

i

)

)

for all

⁢

i

∈

I

.

One readily checks that

ω

A

is a well-defined

n

-ary operator on

A

.

A

equipped with all

ω

A

on

A

is an

O

-algebra, and is called the

direct product

of

A

i

. Each

A

i

is called a

direct factor

of

A

.

If each

A

i

=

B

, where

B

is an

O

-algebra, then we call

A

the direct power of

B

and we write

A

as

B

I

(keep in mind the

isomorphic

identifications).

If

A

is the direct product of

A

i

, then for each

i

∈

I

we can

associate

a

homomorphism

π

i

:

A

→

A

i

called a

projection

given by

π

i

⁢

(

a

)

=

a

⁢

(

i

)

. It is a homomorphism because

π

i

⁢

(

ω

A

⁢

(

a

1

,

…

,

a

n

)

)

=

ω

A

⁢

(

a

1

,

…

,

a

n

)

⁢

(

i

)

=

ω

A

i

⁢

(

a

1

⁢

(

i

)

,

…

,

a

n

⁢

(

i

)

)

=

ω

A

i

⁢

(

π

i

⁢

(

a

1

)

,

…

,

π

i

⁢

(

a

n

)

)

.

Remark

. The direct product of a single algebraic system is the algebraic system itself. An

empty direct product

is defined to be a trivial algebraic system (one-element algebra).

Title

direct product of algebras

Canonical name

DirectProductOfAlgebras

Date of creation

2013-03-22 16:44:35

Last modified on

2013-03-22 16:44:35

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

9

Author

CWoo (3771)

Entry type

Definition

Classification

msc 08A05

Classification

msc 08A62

Defines

direct product

Defines

direct factor

Defines

direct power

Defines

projection

Defines

empty direct product

Generated on Fri Feb 9 14:00:07 2018 by

LaTeXML
isomorphism theorems on algebraic systems

isomorphism theorems on algebraic systems

In this entry, all

algebraic systems

are of the same type; they are all

O

-

algebras

. We list the

generalizations

of three famous isomorphism theorems, familiar to those who have studied

abstract algebra

in college.

Theorem 1

.

If

f

:

A

→

B

is a

homomorphism

from algebras

A

and

B

. Then

A

/

ker

⁡

(

f

)

≅

f

⁢

(

A

)

.

Theorem 2

.

If

B

⊆

A

are algebras and

C

is a

congruence

(

http://planetmath.org/CongruenceRelationOnAnAlgebraicSystem

) on

A

, then

B

/

ℭ

B

≅

B

ℭ

/

ℭ

,

where

C

B

is the congruence restricted to

B

, and

B

C

is the

extension

of

B

by

C

.

Theorem 3

.

If

A

is an algebra and

C

⊆

D

are congruences on

A

. Then

1.

there is a unique homomorphism

f

:

A

/

ℭ

→

A

/

𝔇

such that

\xymatrix

⁢

&

⁢

A

⁢

\ar

⁢

[

d

⁢

l

]

[

⋅

]

ℭ

⁢

\ar

⁢

[

d

⁢

r

]

[

⋅

]

𝔇

⁢

&

⁢

A

/

ℭ

⁢

\ar

⁢

[

r

⁢

r

]

f

⁢

&

⁢

&

⁢

A

/

𝔇

where the downward pointing arrows are the

natural projections

of

A

onto the quotient algebras (induced by the respective congruences).

2.

Furthermore, if

k

⁢

e

⁢

r

⁢

(

f

)

=

𝔇

/

ℭ

, then

–

𝔇

/

ℭ

is a congruence on

A

/

ℭ

, and

–

there is a unique

isomorphism

f

′

:

A

/

ℭ

→

(

A

/

ℭ

)

/

(

𝔇

/

ℭ

)

satisfying the

equation

f

=

[

⋅

]

𝔇

/

ℭ

∘

f

′

. In other words,

(

A

/

ℭ

)

/

(

𝔇

/

ℭ

)

≅

A

/

𝔇

.

Title

isomorphism theorems on algebraic systems

Canonical name

IsomorphismTheoremsOnAlgebraicSystems

Date of creation

2013-03-22 16:45:28

Last modified on

2013-03-22 16:45:28

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

8

Author

CWoo (3771)

Entry type

Theorem

Classification

msc 08A05

Generated on Fri Feb 9 14:00:35 2018 by

LaTeXML
ground fields and rings

ground fields and rings

The following is a list of common uses of the

ground

or

base

field or ring in

algebra

. These are endowed with based on their

context

so the following list may be or may not apply uniformly.

One commonality is generally found for the use of ground ring or field: the result is a unitial subring of the original. Outside of this requirement, the constraints are specific to context.

•

Given a ring

R

with a 1, let

ℤ

⁢

1

be the

subgroup

of

R

generated by

1

under

addition

. This is consequently a subring of

R

of the same

characteristic

as

R

. Thus is it

isomorphic

to

ℤ

/

c

⁢

ℤ

where

c

is the characteristic of

R

. This is the smallest unital subring of

R

and so rightfully may be called the ground or base ring of

R

.

When the characteristic of

R

is prime,

ℤ

⁢

1

≅

ℤ

/

p

⁢

ℤ

and so it may be called the ground field of

R

.

•

Given a

vector space

or algebra

A

over a field

k

, then

k

is the ground/

base field

of

A

.

•

Given a set of matrices

M

n

⁢

(

R

)

, the ground ring is commonly the ring

R

, and if required as a subring of

M

n

⁢

(

R

)

then it is taken as the set of all

scalar matrices

.

•

Given a field extension

K

/

k

over a field

k

, then

k

is the ground field of

K

in this context. For a general field where no specific

subfield

has been specified, the ground/base field then typically defaults to the

prime subfield

of

K

. (Recall the prime subfield is the unique smallest subfield of

K

.)

•

Given a field

K

and a set of

field automorphisms

f

:

K

→

K

,  the ground/base field in this context is the

fixed field

(

http://planetmath.org/Fixed

) of the

automorphisms

. That is, the largest subfield of

K

which is

pointwise

fixed by each

f

. Since a field automorphism must fix the prime subfield, this definition always produces a field containing the prime subfield.

Title

ground fields and rings

Canonical name

GroundFieldsAndRings

Date of creation

2013-03-22 15:54:22

Last modified on

2013-03-22 15:54:22

Owner

Algeboy (12884)

Last modified by

Algeboy (12884)

Numerical id

15

Author

Algeboy (12884)

Entry type

Definition

Classification

msc 08A30

Related topic

ExtensionField

Related topic

FieldAdjunction

Related topic

RingAdjunction

Defines

ground field

Defines

base field

Defines

ground ring

Defines

base ring

Generated on Fri Feb 9 14:01:32 2018 by

LaTeXML
simple algebraic system

simple algebraic system

An

algebraic system

A

is

simple

if the only

congruences

on it are

A

×

A

and

Δ

, the

diagonal relation

.

For example, let’s find out what are the

simple algebras

in the class of groups. Let

G

be a group that is simple in the sense defined above.

First, what are the congruences on

G

? A congruence

C

on

G

is a

subgroup

of

G

×

G

and an

equivalence relation

on

G

at the same time. As an equivalence relation,

C

corresponds to a

partition

of

G

in the following manner:

G

=

⋃

i

∈

I

N

i

and

C

=

⋃

i

∈

I

N

i

2

, where

N

i

∩

N

j

=

∅

for

i

≠

j

. Each of the

N

i

is an

equivalence class

of

C

. Let

N

be the equivalence class containing

1

. If

a

,

b

∈

N

, then

[

a

]

=

[

b

]

=

[

1

]

, so that

[

a

⁢

b

]

=

[

a

]

⁢

[

b

]

=

[

1

]

⁢

[

1

]

=

[

1

]

, or

a

⁢

b

∈

N

. In

addition

,

[

a

-

1

]

=

[

1

]

⁢

[

a

-

1

]

=

[

a

]

⁢

[

a

-

1

]

=

[

a

⁢

a

-

1

]

=

[

1

]

, so

a

1

∈

N

.

N

is a subgroup of

G

. Furthermore, if

c

∈

G

,

[

c

⁢

a

⁢

c

-

1

]

=

[

c

]

⁢

[

a

]

⁢

[

c

-

1

]

=

[

c

]

⁢

[

1

]

⁢

[

c

-

1

]

=

[

c

⁢

c

-

1

]

=

[

1

]

, so that

c

⁢

a

⁢

c

-

1

∈

N

,

N

is a

normal subgroup

of

G

.

Conversely

, given a normal subgroup

N

of

G

, forming left (right) cosets

N

i

of

N

, and taking

C

=

⋃

i

∈

I

N

i

2

gives us the congruence

C

on

G

.

Now, if

G

is simple, then this says that the only congruences on

G

are

G

×

G

and

Δ

, which corresponds to

G

having

G

and

⟨

1

⟩

as the only normal subgroups. So,

G

as a simple algebra is just a

simple group

. Conversely, if

G

is a simple group, then the only congruences on

G

are those corresponding to

G

and

⟨

1

⟩

, the only normal subgroups of

G

. Therefore, a simple group is a simple algebra.

Remark

. Any simple algebraic system is

subdirectly irreducible

.

Title

simple algebraic system

Canonical name

SimpleAlgebraicSystem

Date of creation

2013-03-22 16:46:56

Last modified on

2013-03-22 16:46:56

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

5

Author

CWoo (3771)

Entry type

Definition

Classification

msc 08A30

Synonym

simple

Defines

simple algebra

Generated on Fri Feb 9 14:01:53 2018 by

LaTeXML
coefficients of a polynomial

coefficients of a polynomial

If

p

⁢

(

x

)

=

∑

i

=

0

n

a

i

⁢

x

i

is a

polynomial

(

http://planetmath.org/BasicPolynomial

), then its

coefficients

are

{

a

i

}

i

=

0

n

.

Title

coefficients of a polynomial

Canonical

name

CoefficientsOfAPolynomial

Date of creation

2013-03-22 14:01:06

Last modified on

2013-03-22 14:01:06

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

9

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 08A40

Generated on Fri Feb 9 14:02:08 2018 by

LaTeXML
decomposable curve

decomposable curve

An

algebraic curve

f

⁢

(

x

,

y

)

=

0

is

decomposable

, if the

polynomial

f

⁢

(

x

,

y

)

is in

ℝ

⁢

[

x

,

y

]

; that is, if there are polynomials

g

⁢

(

x

,

y

)

and

h

⁢

(

x

,

y

)

with positive degree in

ℝ

⁢

[

x

,

y

]

such that

f

⁢

(

x

,

y

)

=

g

⁢

(

x

,

y

)

⁢

h

⁢

(

x

,

y

)

.

Example.

The

quadratic curve

x

2

a

2

-

y

2

b

2

=

0

(1)

is decomposable, since the

equation

may be written

(

x

a

+

y

b

)

⁢

(

x

a

-

y

b

)

=

0

or equivalently

x

a

+

y

b

=

0

∨

x

a

-

y

b

=

0

.

Thus the curve (1) consists of two

intersecting lines

.

Analogically, one can say that an algebraic surface

g

⁢

(

x

,

y

,

z

)

=

0

is

decomposable

, e.g.

(

x

+

y

+

z

)

2

-

1

=

0

which consists of two

parallel planes

.

Title

decomposable curve

Canonical name

DecomposableCurve

Date of creation

2013-03-22 19:19:38

Last modified on

2013-03-22 19:19:38

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

7

Author

pahio (2872)

Entry type

Definition

Classification

msc 08A40

Classification

msc 26A09

Related topic

Hyperbola2

Defines

decomposable

Defines

decomposable surface

Generated on Fri Feb 9 14:02:10 2018 by

LaTeXML
discriminator function

discriminator function

Let

A

be a non-empty set. The

ternary discriminator

on

A

is the ternary operation

t

on

A

such that

t

⁢

(

a

,

b

,

c

)

:=

{

a

if

⁢

a

≠

b

,

c

otherwise.

In other words,

t

is a

function

that determines whether or not a pair of

elements

in

A

are the same, hence the name discriminator.

It is easy to see that, by setting two of the three

variables

the same,

t

becomes a

constant function

:

t

⁢

(

a

,

b

,

a

)

=

a

,

t

⁢

(

a

,

a

,

b

)

=

b

, and

t

⁢

(

a

,

b

,

b

)

=

a

.

More generally, the

quaternary discriminator

or the

switching function

on

A

is the quaternary

operation

q

on

A

such that

q

⁢

(

a

,

b

,

c

,

d

)

:=

{

d

if

⁢

a

≠

b

,

c

otherwise.

However, this

generalization

is really an

equivalent

concept

in the sense that one can derive one type of discriminator from another: given

q

above, set

t

⁢

(

a

,

b

,

c

)

=

q

⁢

(

a

,

b

,

c

,

a

)

.

Conversely

, given

t

above, set

q

⁢

(

a

,

b

,

c

,

d

)

=

t

⁢

(

t

⁢

(

a

,

b

,

c

)

,

t

⁢

(

a

,

b

,

d

)

,

d

)

.

Remark

. The following ternary functions

t

1

,

t

2

:

A

3

→

A

could also serve as discriminator functions:

t

1

⁢

(

a

,

b

,

c

)

:=

{

b

if

⁢

a

≠

b

,

c

otherwise.

t

2

⁢

(

a

,

b

,

c

)

:=

{

c

if

⁢

a

≠

b

,

a

otherwise.

But they are really no different from

the

ternary discriminator

t

:

t

1

⁢

(

a

,

b

,

c

)

=

t

⁢

(

b

,

a

,

c

)

and

t

⁢

(

a

,

b

,

c

)

=

t

1

⁢

(

b

,

a

,

c

)

,

t

2

⁢

(

a

,

b

,

c

)

=

t

⁢

(

c

,

t

⁢

(

a

,

b

,

c

)

,

a

)

and

t

⁢

(

a

,

b

,

c

)

=

t

2

⁢

(

a

,

t

2

⁢

(

a

,

b

,

c

)

,

c

)

.

References

1

G. Grätzer:

Universal Algebra

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

2

S. Burris, H.P. Sankappanavar:

A Course in Universal Algebra

, Springer, New York (1981).

Title

discriminator function

Canonical name

DiscriminatorFunction

Date of creation

2013-03-22 18:20:58

Last modified on

2013-03-22 18:20:58

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

6

Author

CWoo (3771)

Entry type

Definition

Classification

msc 08A40

Synonym

switching function

Defines

ternary discriminator

Defines

quaternary discriminator

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subfunction

subfunction

Definition

. Let

f

:

A

→

B

and

g

:

C

→

D

be

partial functions

.

g

is said to be a

subfunction

of

f

if

g

⊆

f

∩

(

C

×

D

)

.

In other words,

g

is a subfunction of

f

iff whenever

x

∈

C

such that

g

⁢

(

x

)

is defined, then

x

∈

A

,

f

⁢

(

x

)

is defined, and

g

⁢

(

x

)

=

f

⁢

(

x

)

.

If we set

C

′

=

A

∩

C

and

D

′

=

B

∩

D

, then

g

⊆

f

∩

(

C

′

×

D

′

)

, so there is no harm in assuming that

C

and

D

are subsets of

A

and

B

respectively, which we will do for the rest of the discussion.

In practice, whenever

g

is a subfunction of

f

, we often assume that

g

and

f

have the same domain and

codomain

. Otherwise, we would specify that

g

is a subfunction of

f

:

A

→

B

with domain

C

and codomain

D

.

For example,

f

:

ℝ

→

ℝ

defined by

f

⁢

(

x

)

=

x

2

-

1

is a partial function, whose domain of definition is

(

-

∞

,

-

1

]

∪

[

1

,

∞

)

, and the partial function

g

:

ℝ

→

ℝ

given by

g

⁢

(

x

)

=

x

2

-

1

x

2

-

1

is a subfunction of

f

. The domain of definition of

g

is

(

-

∞

,

-

1

)

∪

(

1

,

∞

)

.

Two immediate properties of a subfunction

g

:

C

→

D

of

f

:

A

→

B

are

•

the range of

g

is a subset of the range of

f

:

g

⁢

(

C

)

⊆

f

⁢

(

C

)

,

•

the domain of definition of

g

is a subset of the domain of definition of

f

:

g

-

1

⁢

(

D

)

⊆

f

-

1

⁢

(

D

)

.

Definition

. A subfunction

g

:

C

→

D

of

f

:

A

→

B

is called a

restriction

of

f

relative to

D

, if

g

⁢

(

C

)

=

f

⁢

(

C

)

∩

D

, and a

restriction of

f

if

g

⁢

(

C

)

=

f

⁢

(

C

)

.

Every partial function

g

:

C

→

D

corresponds to a unique restriction

g

′

:

C

→

g

⁢

(

C

)

of

g

.

A restriction

g

:

C

→

D

of

f

:

A

→

B

is certainly a restriction of

f

relative to

D

, since

f

⁢

(

C

)

∩

D

=

g

⁢

(

C

)

∩

D

=

g

⁢

(

C

)

, but not

conversely

. For example, let

A

be the set of all non-negative integers and

-

A

:

A

2

→

A

the ordinary

subtraction

.

-

A

is easily seen to be a partial function. Let

B

be the

set of all positive integers

. Then

-

B

:

B

2

→

B

is a restriction of

-

A

:

A

2

→

A

, relative to

B

. However,

-

B

is not a restriction of

-

A

, for

n

-

B

n

is not defined, while

n

-

A

n

=

0

∈

A

.

References

1

G. Grätzer:

Universal Algebra

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

Title

subfunction

Canonical name

Subfunction

Date of creation

2013-03-22 18:41:54

Last modified on

2013-03-22 18:41:54

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

8

Author

CWoo (3771)

Entry type

Definition

Classification

msc 08A55

Classification

msc 03E20

Defines

restriction

Generated on Fri Feb 9 14:03:43 2018 by

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subdirect product of algebraic systems

subdirect product of algebraic systems

In this entry, all

algebraic systems

are of the same type. For each algebraic system, we drop the associated operator set for simplicity.

Let

A

i

be algebraic systems

indexed by

i

∈

I

.

B

is called a

subdirect product

of

A

i

if

1.

B

is a

subalgebra

of the

direct product

of

A

i

.

2.

for each

i

∈

I

,

π

i

⁢

(

B

)

=

A

i

.

In the second condition,

π

i

denotes the

projection

homomorphism

∏

A

i

→

A

i

. By

restriction

, we may consider

π

i

as homomorphisms

B

→

A

i

. When

B

is

isomorphic

to

∏

A

i

, then

B

is a

trivial subdirect product

of

A

i

.

This generalizes the notion of a direct product, since in many

instances

, an algebraic system can not be decomposed into a direct product of algebras.

When all

A

i

=

C

for some algebraic system

C

of the same type, then

B

is called a

subdirect power

of

C

.

Remarks

.

1.

A very simple example of a subdirect product is the following: let

A

1

=

A

2

=

{

1

,

2

,

3

}

. Then the subset

B

=

{

(

x

,

y

)

∈

A

1

×

A

2

∣

x

≤

y

}

is a subdirect product of the sets

A

1

and

A

2

(considered as algebraic systems with no

operators

).

2.

Let

B

is a subdirect product of

A

i

, and

p

i

:=

(

π

i

)

B

, the restriction of

π

i

to

B

. Then

B

/

ker

⁡

(

p

i

)

≅

A

i

. In

addition

,

⋂

{

ker

⁡

(

p

i

)

∣

i

∈

I

}

=

Δ

,

where

Δ

is the

diagonal relation

. To see the last equality, suppose

a

,

b

∈

B

with

a

≡

b

(

mod

p

i

)

. Then

a

⁢

(

i

)

=

π

i

⁢

(

a

)

=

p

i

⁢

(

a

)

=

p

i

⁢

(

b

)

=

π

i

⁢

(

b

)

=

b

⁢

(

i

)

. Since this is true for every

i

∈

I

,

a

=

b

.

3.

Conversely

, if

A

is an algebraic system and

{

ℭ

i

∣

i

∈

I

}

is a set of

congruences

on

A

such that

⋂

{

ℭ

i

∣

i

∈

I

}

=

Δ

.

Then

A

is isomorphic to a subdirect product of

A

/

ℭ

i

.

4.

An algebraic system is said to be

subdirectly irreducible

if, whenever

ℭ

i

are congruences on

A

and

⋂

{

ℭ

i

∣

i

∈

I

}

=

Δ

, then one of

ℭ

i

=

Δ

.

5.

Birkhoff’s

Theorem

on the Decomposition of an Algebraic System

. Every algebraic system is isomorphic to a subdirect product of subdirectly irreducible algebraic systems. This works only when the algebraic system is finitary.

Title

subdirect product of algebraic systems

Canonical name

SubdirectProductOfAlgebraicSystems

Date of creation

2013-03-22 16:44:51

Last modified on

2013-03-22 16:44:51

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

10

Author

CWoo (3771)

Entry type

Definition

Classification

msc 08A62

Classification

msc 08A05

Classification

msc 08B26

Defines

subdirect product

Defines

subdirect power

Defines

subdirectly irreducible

Defines

trivial subdirect product

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