factor embeddable

factor embeddable

Let

K

be a class of models (

structures

) of a given

signature

. Consider a (non-empty) family of structures

{

A

i

:

i

∈

I

}

in

K

. If

j

∈

I

and

f

:

A

j

→

∏

i

∈

I

A

i

is an

embedding

, we say that

f

is a

factor embedding

.

[

1

,

2

]

If additionally

f

satisfies

the condition that

π

j

∘

f

is the

identity

on

A

i

, where

π

j

:

∏

i

∈

I

A

i

→

A

j

is the

j

th

projection

, then

f

is said to be a

strong factor embedding

.

[

2

]

K

is said to be a

factor embeddable

class iff for every (non-empty) family of structures

{

A

i

:

i

∈

I

}

in

K

and every

j

∈

I

there is a factor embedding

f

:

A

j

→

∏

i

∈

I

A

i

.

[

1

,

2

]

The definition above does not require the

product

∏

i

∈

I

A

i

to be a member of

K

, however many interesting examples of factor embeddable classes are in fact

closed under

products. Factor embeddable classes that are closed under finite products (or equivalently under binary products) have the

joint embedding property

. Factor embeddable classes closed under arbitrary products have the strong joint embedding property.

0.0.1

Characterization

Factor embeddable classes have an easy to prove but somewhat unintuitive

characterization

which does not mention the

concepts

of product or embedding:

The following are

equivalent

for a class

K

of models

[

2

]

:

1.

K

is factor embeddable.

2.

For every pair of models

A

,

B

∈

K

there exists a

homomorphism

f

:

A

→

B

.

To see the above, suppose

K

is factor embeddable and consider models

A

,

B

∈

K

. Then there exists a factor embedding from

A

into the product

A

×

B

. Composing this embedding with the projection onto

B

gives a homomorphism

f

:

A

→

B

.

Conversely

suppose such a homomorphism

f

:

A

→

B

exists for all

A

,

B

∈

K

and consider a family

{

A

i

:

i

∈

I

}

in

K

and

j

∈

I

. We can define a strong factor embedding

f

:

A

j

→

∏

i

∈

I

A

i

by choosing homomorphisms

f

i

:

A

j

→

A

i

for all

i

∈

I

with

f

j

the

identity map

on

A

j

, and then for all

a

∈

A

setting

f

⁢

(

a

)

i

=

f

i

⁢

(

a

)

for each

i

∈

I

.

[

2

]

The above proof shows that the factor embeddings guaranteed to exist for a factor embeddable class can always be chosen to be strong factor emebeddings.

[

2

]

A corollory of the above is that if there exists a model which is a retract of every member of a class

K

then,

K

is factor embeddable - in particular if the members of

K

have one

element

submodels

, then

K

is factor embeddable.

[

2

]

(A retract of a model is a submodel which is also a

quotient

model such that the quotient map composed with the submodel embedding is the identity map.)

0.0.2

Examples

The following are examples of factor embeddable classes:

•

The

variety

of all groups (the

trivial group

is a one element

subalgebra

of every group)

•

The variety of all lattices (every

lattice

has one element sublattices)

•

The class of all non-trivial

Boolean algebras

(the two element Boolean algebra is a retract of all non-trivial Boolean algebras)

The class of all Boolean algebras is an example of a class which is not factor embeddable - there is no way to embed the

trivial

Boolean algebra into a product of itself with any non-trivial Boolean algebras. (The trivial Boolean algebra satisfies the identity

0

=

1

which is not satisfied by any Boolean algebra having more than one element.)

References

1

Peter Bruyns, Henry Rose:

Varieties with

cofinal

sets: examples and amalgamation

, Proc. Amer. Math. Soc. 111 (1991), 833-840

2

Colin Naturman, Henry Rose:

Ultra-universal models

, Quaestiones Mathematicae, 15(2), 1992, 189-195

Title

factor embeddable

Canonical name

FactorEmbeddable

Date of creation

2013-03-22 19:36:55

Last modified on

2013-03-22 19:36:55

Owner

Naturman (26369)

Last modified by

Naturman (26369)

Numerical id

19

Author

Naturman (26369)

Entry type

Definition

Classification

msc 03C52

Related topic

JointEmbeddingProperty

Defines

factor embeddable class

Defines

factor embedding

Defines

strong factor embedding

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