aarjaneiro/planetmath huggingface dataset:
every congruence is the kernel of a homomorphism

every congruence is the kernel of a homomorphism

Let

Σ

be a fixed

signature

, and

𝔄

a

structure

for

Σ

. If

∼

is a

congruence

on

𝔄

, then there is a

homomorphism

f

such that

∼

is the kernel of

f

.

Proof.

Define a homomorphism

f

:

𝔄

→

𝔄

/

∼

:

a

↦

[

[

a

]

]

. Observe that

a

∼

b

if and only if

f

⁢

(

a

)

=

f

⁢

(

b

)

, so

∼

is the kernel of

f

. To verify that

f

is a homomorphism, observe that

1.

For each constant symbol

c

of

Σ

,

f

⁢

(

c

𝔄

)

=

[

[

c

𝔄

]

]

=

c

𝔄

⁣

/

∼

.

2.

For each

n

∈

ℕ

and each

n

-ary function symbol

F

of

Σ

,

f

⁢

(

F

𝔄

⁢

(

a

1

,

…

⁢

a

n

)

)

=

[

[

F

𝔄

⁢

(

a

1

,

…

⁢

a

n

)

]

]

=

F

𝔄

⁣

/

∼

⁢

(

[

[

a

1

]

]

,

…

⁢

[

[

a

n

]

]

)

=

F

𝔄

⁣

/

∼

⁢

(

f

⁢

(

a

1

)

,

…

⁢

f

⁢

(

a

n

)

)

.

∎

3.

For each

n

∈

ℕ

and each

n

-ary relation symbol

R

of

Σ

,
if

R

𝔄

⁢

(

a

1

,

…

,

a

n

)

then

R

𝔄

⁣

/

∼

⁢

(

[

[

a

1

]

]

,

…

,

[

[

a

n

]

]

)

, so

R

𝔄

⁣

/

∼

⁢

(

f

⁢

(

a

1

)

,

…

,

f

⁢

(

a

n

)

)

.

Title

every congruence is the kernel of a homomorphism

Canonical name

EveryCongruenceIsTheKernelOfAHomomorphism

Date of creation

2013-03-22 13:48:59

Last modified on

2013-03-22 13:48:59

Owner

almann (2526)

Last modified by

almann (2526)

Numerical id

11

Author

almann (2526)

Entry type

Theorem

Classification

msc 03C07

Classification

msc 03C05

Generated on Thu Feb 8 19:54:44 2018 by

LaTeXML
example of definable type

example of definable type

Consider

(

𝐐

,

<

)

as a

structure

in a

language

with one

binary relation

, which we interpret as the order.
This is a

universal

,

ℵ

0

-categorical structure (see example of universal structure).

The theory of

(

𝐐

,

<

)

has

quantifier elimination

, and so is

o-minimal

.
Thus a type over the set

𝐐

is determined by the

quantifier free formulas

over

𝐐

, which in turn are determined by the atomic formulas over

𝐐

.
An atomic formula in one

variable

over

B

is of the form

x

<

b

or

x

>

b

or

x

=

b

for some

b

∈

B

.
Thus each 1-type over

𝐐

determines a

Dedekind cut

over

𝐐

, and

conversely

a Dedekind cut determines a

complete type

over

𝐐

.
Let

D

⁢

(

p

)

:=

{

a

∈

𝐐

:

x

>

a

∈

p

}

.

Thus there are two classes of type over

𝐐

.

1.

Ones where

D

⁢

(

p

)

is of the form

(

-

∞

,

a

)

or

(

-

∞

,

a

]

for some

a

∈

𝐐

. It is clear that these are

definable

from the above discussion.

2.

Ones where

D

⁢

(

p

)

has no

supremum

in

𝐐

. These are clearly not definable by o-minimality of

𝐐

.

Title

example of

definable type

Canonical name

ExampleOfDefinableType

Date of creation

2013-03-22 13:29:43

Last modified on

2013-03-22 13:29:43

Owner

aplant (12431)

Last modified by

aplant (12431)

Numerical id

5

Author

aplant (12431)

Entry type

Example

Classification

msc 03C07

Related topic

ExampleOfUniversalStructure

Related topic

DedekindCuts

Generated on Thu Feb 8 19:54:46 2018 by

LaTeXML
first isomorphism theorem

first isomorphism theorem

Let

Σ

be a fixed

signature

, and

𝔄

and

𝔅

structures

for

Σ

. If

f

:

𝔄

→

𝔅

is a

homomorphism

, then there is a unique

bimorphism

ϕ

:

𝔄

/

ker

⁡

(

f

)

→

im

⁡

(

f

)

such that for all

a

∈

𝔄

,

ϕ

⁢

(

[

[

a

]

]

)

=

f

⁢

(

a

)

. Furthermore, if

f

has the additional property that for each

n

∈

ℕ

and each

n

-ary relation symbol

R

of

Σ

,

R

𝔅

(

f

(

a

1

)

,

…

,

f

(

a

n

)

)

⇒

∃

a

i

′

[

f

(

a

i

)

=

f

(

a

i

′

)

∧

R

𝔄

(

a

1

′

,

…

,

a

n

′

)

]

,

then

ϕ

is an

isomorphism

.

Proof.

Since the homomorphic image of a

Σ

-structure is also a

Σ

-structure, we may assume that

im

⁡

(

f

)

=

𝔅

.

Let

∼

=

ker

(

f

)

. Define a bimorphism

ϕ

:

𝔄

/

∼

→

𝔅

:

[

[

a

]

]

↦

f

(

a

)

. To verify that

ϕ

is well defined, let

a

∼

a

′

. Then

ϕ

⁢

(

[

[

a

]

]

)

=

f

⁢

(

a

)

=

f

⁢

(

a

′

)

=

ϕ

⁢

(

[

[

a

′

]

]

)

. To show that

ϕ

is

injective

, suppose

ϕ

⁢

(

[

[

a

]

]

)

=

ϕ

⁢

(

[

[

a

′

]

]

)

. Then

f

⁢

(

a

)

=

f

⁢

(

a

′

)

, so

a

∼

a

′

. Hence

[

[

a

]

]

=

[

[

a

′

]

]

. To show that

ϕ

is a homomorphism, observe that for any constant symbol

c

of

Σ

we have

ϕ

⁢

(

[

[

c

𝔄

]

]

)

=

f

⁢

(

c

𝔄

)

=

c

𝔅

. For each

n

∈

ℕ

and each

n

-ary function symbol

F

of

Σ

,

ϕ

⁢

(

F

𝔄

⁣

/

∼

⁢

(

[

[

a

1

]

]

,

…

,

[

[

a

n

]

]

)

)

=

ϕ

⁢

(

[

[

F

𝔄

⁢

(

a

1

,

…

,

a

n

)

]

]

)

=

f

⁢

(

F

𝔄

⁢

(

a

1

,

…

,

a

n

)

)

=

F

𝔅

⁢

(

f

⁢

(

a

1

)

,

…

,

f

⁢

(

a

n

)

)

=

F

𝔅

(

ϕ

(

[

[

a

1

]

]

,

…

,

ϕ

(

[

[

a

n

]

]

)

)

.

For each

n

∈

ℕ

and each

n

-ary relation symbol

R

of

Σ

,

R

𝔄

⁣

/

∼

⁢

(

[

[

a

1

]

]

,

…

,

[

[

a

n

]

]

)

⇒

R

𝔄

⁢

(

a

1

,

…

,

a

n

)

⇒

R

𝔅

⁢

(

f

⁢

(

a

1

)

,

…

,

f

⁢

(

a

n

)

)

⇒

R

𝔅

(

ϕ

(

[

[

a

1

]

]

,

…

,

ϕ

(

[

[

a

n

]

]

)

)

.

Thus

ϕ

is a bimorphism.

Now suppose

f

has the additional property mentioned in the statement of the

theorem

. Then

R

𝔅

⁢

(

ϕ

⁢

(

[

[

a

1

]

]

)

,

…

,

ϕ

⁢

(

[

[

a

n

]

]

)

)

⇒

R

𝔅

⁢

(

f

⁢

(

a

1

)

,

…

,

f

⁢

(

a

n

)

)

⇒

∃

a

i

′

[

a

i

∼

a

i

′

∧

R

𝔄

(

a

1

′

,

…

,

a

n

′

)

]

⇒

R

𝔄

⁣

/

∼

⁢

(

[

[

a

1

]

]

,

…

,

[

[

a

n

]

]

)

.

Thus

ϕ

is an isomorphism.
∎

Title

first isomorphism theorem

Canonical name

FirstIsomorphismTheorem1

Date of creation

2013-03-22 13:50:42

Last modified on

2013-03-22 13:50:42

Owner

almann (2526)

Last modified by

almann (2526)

Numerical id

10

Author

almann (2526)

Entry type

Theorem

Classification

msc 03C07

Generated on Thu Feb 8 19:54:50 2018 by

LaTeXML
first-order theory

first-order theory

In what follows, references to

sentences

and sets of sentences are
all relative to some fixed

first-order language

L

.

Definition.

A

theory

T

is a

deductively
closed

set of sentences in

L

; that is, a set

T

such that for each
sentence

φ

,

T

⊢

φ

only if

φ

∈

T

.

Remark

. Some authors do not require that a theory be deductively closed. Therefore, a theory is simply a set of sentences. This is not a cause for alarm, since every theory

T

under this definition can be “extended” to a deductively closed theory

T

⊢

:=

{

φ

∈

L

∣

T

⊢

φ

}

. Furthermore,

T

⊢

is unique (it is the smallest deductively closed theory including

T

), and any

structure

M

is a model of

T

iff it is a model of

T

⊢

.

Definition.

A theory

T

is

consistent

if and only
if for some sentence

φ

,

T

⊬

φ

.
Otherwise,

T

is

inconsistent

. A sentence

φ

is

consistent with

T

if and only if the
theory

T

∪

{

φ

}

is consistent.

Definition.

A theory

T

is

complete

if and only
if

T

is consistent and for each sentence

φ

, either

φ

∈

T

or

¬

⁢

φ

∈

T

.

Lemma.

A consistent theory

T

is complete if and only if

T

is
maximally consistent. That is,

T

is complete if and only if for
each sentence

φ

,

φ

∉

T

only if

T

∪

{

φ

}

is inconsistent. See this entry (

http://planetmath.org/MaximallyConsistent

) for a proof.

Theorem

. (Tarski)

Every consistent theory

T

is included
in a complete theory.

Proof :

Use Zorn’s lemma on the set of consistent
theories that include

T

.

Remark

. A theory

T

is

axiomatizable

if and only
if

T

includes a

decidable

(

http://planetmath.org/DecidableSet

) subset

Δ

such that

Δ

⊢

T

(every sentence of

T

is a

logical consequence

of

Δ

), and

finitely axiomatizable

if

Δ

can be made finite. Every complete axiomatizable theory

T

is decidable;
that is, there is an algorithm that given a sentence

φ

as
input yields

0

if

φ

∈

T

, and

1

otherwise.

Title

first-order theory

Canonical name

FirstorderTheory

Date of creation

2013-03-22 12:43:04

Last modified on

2013-03-22 12:43:04

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

19

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03C07

Classification

msc 03B10

Synonym

first order theory

Related topic

PropertiesOfConsistency

Related topic

MaximallyConsistent

Defines

theory

Defines

complete theory

Defines

axiomatizable theory

Defines

deductively closed

Defines

finitely axiomatizable theory

Generated on Thu Feb 8 19:55:03 2018 by

LaTeXML
length of a string

length of a string

Suppose we have a string

w

on

alphabet

Σ

. We can then represent
the string as

w

=

x

1

⁢

x

2

⁢

x

3

⁢

⋯

⁢

x

n

-

1

⁢

x

n

, where for all

x

i

(

1

≤

i

≤

n

),

x

i

∈

Σ

(this means that each

x

i

must
be a “letter” from the alphabet). Then, the length of

w

is

n

. The

length of a string

w

is represented as

∥

w

∥

.

For example, if our alphabet is

Σ

=

{

a

,

b

,

c

⁢

a

}

then the length of the string

w

=

b

⁢

c

⁢

a

⁢

a

⁢

b

is

∥

w

∥

=

4

, since
the string breaks down as follows:

x

1

=

b

,

x

2

=

c

⁢

a

,

x

3

=

a

,

x

4

=

b

.
So, our

x

n

is

x

4

and therefore

n

=

4

. Although you may think that

c

⁢

a

is two separate symbols, our chosen alphabet in fact classifies it as
a single symbol.

A “special case” occurs when

∥

w

∥

=

0

, i.e. it does not have any symbols
in it. This string is called the

empty string

. Instead of saying

w

=

, we use

λ

to represent the empty string:

w

=

λ

. This is

similar

to the practice of using

β

to represent a space, even though a space is really blank.

If your alphabet contains

λ

as a symbol, then
you must use something else to denote the empty string.

Suppose you also have a string

v

on the same alphabet as

w

. We turn

w

into

x

1

⁢

⋯

⁢

x

n

just as before, and similarly

v

=

y

1

⁢

⋯

⁢

y

m

.
We say

v

is

equal

to

w

if and only if both

m

=

n

, and
for every

i

,

x

i

=

y

i

.

For example, suppose

w

=

b

⁢

b

⁢

a

and

v

=

b

⁢

a

⁢

b

, both strings on alphabet

Σ

=

{

a

,

b

}

.
These strings are not equal because the second symbols do not match.

Title

length of a string

Canonical name

LengthOfAString

Date of creation

2013-03-22 12:29:16

Last modified on

2013-03-22 12:29:16

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

7

Author

mathcam (2727)

Entry type

Definition

Classification

msc 03C07

Defines

equality of strings

Defines

empty string

Generated on Thu Feb 8 19:55:12 2018 by

LaTeXML
proof of homomorphic image of a Σ-structure is a Σ-structure

proof of homomorphic image of a

Σ

-structure is a

Σ

-structure

We need to show that

im

⁡

(

f

)

is

closed under

functions

. For every

constant symbol

c

of

Σ

,

c

𝔅

=

f

⁢

(

c

𝔄

)

. Hence

c

𝔅

∈

im

⁡

(

f

)

. Also, if

b

1

,

…

,

b

n

∈

im

⁡

(

f

)

and

F

is an

n

-ary function symbol of

Σ

, then for some

a

1

,

…

,

a

n

∈

𝔄

we have

F

𝔅

⁢

(

b

1

,

…

,

b

n

)

=

F

𝔅

⁢

(

f

⁢

(

a

1

)

,

…

,

f

⁢

(

a

n

)

)

=

f

⁢

(

F

𝔄

⁢

(

a

1

,

…

,

a

n

)

)

.

Hence

F

𝔅

⁢

(

b

1

,

…

,

b

n

)

∈

im

⁡

(

f

)

.

Title

proof of

homomorphic image

of a

Σ

-

structure

is a

Σ

-structure

Canonical name

ProofOfHomomorphicImageOfASigmastructureIsASigmastructure

Date of creation

2013-03-22 13:46:47

Last modified on

2013-03-22 13:46:47

Owner

almann (2526)

Last modified by

almann (2526)

Numerical id

4

Author

almann (2526)

Entry type

Proof

Classification

msc 03C07

Generated on Thu Feb 8 19:55:19 2018 by

LaTeXML
recursively axiomatizable theory

recursively axiomatizable theory

Let

T

be a

first order theory

. A subset

Δ

⊆

T

is a set of

axioms

for

T

if and only if

T

is the set of all

consequences

of the

formulas

in

Δ

. In other words,

φ

∈

T

if and only if

φ

is provable using only

assumptions

from

Δ

.

Definition.

A theory

T

is said to be

finitely axiomatizable

if and only if there is a

finite set

of axioms for

T

; it is said to be

recursively axiomatizable

if and only if it has a

recursive set

of axioms.

For example,

group theory

is finitely axiomatizable (it has only three axioms), and

Peano arithmetic

is recursivaly axiomatizable : there is clearly an algorithm that can decide if a formula of the

language

of the

natural numbers

is an axiom.

Theorem

.

recursively axiomatizable theories are

decidable

.

As an example of the use of this theorem, consider the theory of algebraically closed fields of

characteristic

p

for any number

p

prime or 0. It is

complete

, and the set of axioms is obviously

recursive

, and so it is decidable.

Title

recursively axiomatizable theory

Canonical name

RecursivelyAxiomatizableTheory

Date of creation

2013-03-22 12:43:13

Last modified on

2013-03-22 12:43:13

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

7

Author

mathcam (2727)

Entry type

Definition

Classification

msc 03C07

Defines

finitely axiomatizable

Generated on Thu Feb 8 19:55:25 2018 by

LaTeXML
satisfaction relation

satisfaction relation

Alfred Tarski was the first mathematician to give a formal definition of what
it means for a

formula

to be “true” in a

structure

. To do this, we
need to provide a meaning to terms, and truth-values to the formulas.
In doing this,

free variables

cause a problem : what value are they
going to have ? One possible answer is to supply temporary values for the free variables, and define our notions in terms of these temporary values.

Let

𝒜

be a structure with

signature

τ

.
Suppose

ℐ

is an

interpretation

, and

σ

is a

function

that
assigns elements of

A

to variables, we define the function

Val

ℐ

,

σ

inductively on the construction of terms :

Val

ℐ

,

σ

⁡

(

c

)

=

ℐ

⁢

(

c

)

c

⁢

a constant symbol

Val

ℐ

,

σ

⁡

(

x

)

=

σ

⁢

(

x

)

x

⁢

a variable

Val

ℐ

,

σ

⁡

(

f

⁢

(

t

1

,

…

,

t

n

)

)

=

ℐ

⁢

(

f

)

⁢

(

Val

ℐ

,

σ

⁡

(

t

1

)

,

…

,

Val

ℐ

,

σ

⁡

(

t

n

)

)

f

⁢

an

⁢

n

⁢

-ary function symbol

Now we are set to define

satisfaction

. Again we have to take care of free variables by assigning temporary values to them via a function

σ

.
We define the

relation

𝒜

,

σ

⊧

φ

by

induction

on the construction of formulas :

𝒜

,

σ

⊧

t

1

=

t

2

⁢

if and only if

⁢

Val

ℐ

,

σ

⁡

(

t

1

)

=

Val

ℐ

,

σ

⁡

(

t

2

)

𝒜

,

σ

⊧

R

⁢

(

t

1

,

…

,

t

n

)

⁢

if and only if

⁢

(

Val

ℐ

,

σ

⁡

(

t

1

)

,

…

,

Val

ℐ

,

σ

⁡

(

t

1

)

)

∈

ℐ

⁢

(

R

)

𝒜

,

σ

⊧

¬

⁢

φ

⁢

if and only if

⁢

𝒜

,

σ

⊧̸

φ

𝒜

,

σ

⊧

φ

∨

ψ

⁢

if and only if either

⁢

𝒜

,

σ

⊧

ψ

⁢

or

⁢

𝔄

,

σ

⊧

ψ

𝒜

,

σ

⊧

∃

x

.

φ

⁢

(

x

)

⁢

if and only if for some

⁢

a

∈

A

,

𝒜

,

σ

⁢

[

x

/

a

]

⊧

φ

Here

σ

⁢

[

x

/

a

]

⁢

(

y

)

⁢

{

a

if

⁢

x

=

y

σ

⁢

(

y

)

else.

In case for some

φ

of

L

, we have

𝒜

,

σ

⊧

φ

, we say that

𝒜

models

, or

is a model of

, or

satisfies

φ

. If

φ

has the free variables

x

1

,

…

,

x

n

, and

a

1

,

…

,

a

n

∈

A

, we also write

𝒜

⊧

φ

⁢

(

a

1

,

…

,

a

n

)

or

𝒜

⊧

φ

⁢

(

a

1

/

x

1

,

…

,

a

n

/

x

n

)

instead of

𝒜

,

σ

⁢

[

x

1

/

a

1

]

⁢

⋯

⁢

[

x

n

/

a

n

]

⊧

φ

. In case

φ

is a

sentence

(formula with no free variables), we write

𝒜

⊧

φ

.

Title

satisfaction relation

Canonical name

SatisfactionRelation

Date of creation

2013-03-22 12:43:56

Last modified on

2013-03-22 12:43:56

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

14

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03C07

Related topic

Model

Defines

satisfaction

Defines

satisfy

Generated on Thu Feb 8 19:55:31 2018 by

LaTeXML
strongly minimal

strongly minimal

Let

L

be a

first order language

and let

M

be an

L

-

structure

. Let

S

, a subset of the domain of

M

be a

definable

infinite set

. Then

S

is

minimal

iff every definable

C

⊆

S

we have either

C

is finite or

S

∖

C

is finite. We say that

M

is

minimal

iff the domain of

M

is a strongly minimal set.

We say that

M

is

strongly minimal

iff for every

N

≡

M

, we have that

N

is minimal. Thus if

T

is a

complete

L

theory then we say

T

is

strongly minimal

if it has some model (equivalently all models) which is strongly minimal.

Note that

M

is strongly minimal iff every definable subset of

M

is

quantifier free

definable in a

language

with just equality. Compare this to the notion of o-minimal structures.

Title

strongly minimal

Canonical name

StronglyMinimal

Date of creation

2013-03-22 13:27:13

Last modified on

2013-03-22 13:27:13

Owner

Timmy (1414)

Last modified by

Timmy (1414)

Numerical id

5

Author

Timmy (1414)

Entry type

Definition

Classification

msc 03C07

Classification

msc 03C10

Classification

msc 03C45

Related topic

OMinimality

Defines

strongly minimal

Defines

minimal

Generated on Thu Feb 8 19:55:36 2018 by

LaTeXML
structure

structure

Let

τ

be a

signature

.
A

τ

-

structure

𝒜

comprises of a set

A

, called the

(or

underlying set

or

) of

𝒜

, and an

interpretation

of the symbols of

τ

as follows:

•

for each constant symbol

c

∈

τ

, an

element

c

A

∈

A

;

•

for each

n

-ary function symbol

f

∈

τ

,
a

function

(or

operation

)

f

A

:

A

n

→

A

;

•

for each

n

-ary relation symbol

R

∈

τ

,
a

n

-ary

relation

R

A

on

A

.

Some authors require that

A

be non-empty.

If

𝒜

is a structure, then the

cardinality

(or

power

) of

𝒜

,

|

𝒜

|

, is the cardinality of its

A

.

Examples of structures abound in mathematics. Here are some of them:

1.

A set is a structure, with no

constants

, no functions, and no relations on it.

2.

A

partially ordered set

is a structure, with one

binary relation

call

partial order

defined on the underlying set.

3.

A group is a structure, with one

binary operation

called

multiplication

, one unary operation called

inverse

, and one constant called the

multiplicative identity

.

4.

A

vector space

is a structure, with one binary operation called

addition

, unary operations called

scalar multiplications

, one for each element of the underlying set, and one constant

0

, the

additive identity

.

5.

A

partially ordered group

is a structure like a group, but with the addition of a partial order on the underlying set.

If

τ

contains only relation symbols, then a

τ

-structure is called a

relational structure

. If

τ

contains only function symbols, then a

τ

-structure is called an

algebraic structure

. In the examples above,

2

is a relation structure, while

3

,

4

are algebraic structures.

Title

structure

Canonical name

Structure

Date of creation

2013-05-20 18:26:21

Last modified on

2013-05-20 18:26:21

Owner

CWoo (3771)

Last modified by

unlord (1)

Numerical id

23

Author

CWoo (1)

Entry type

Definition

Classification

msc 03C07

Related topic

Substructure

Related topic

AlgebraicStructure

Related topic

Model

Related topic

RelationalSystem

Defines

structure

Defines

interpretation

Generated on Thu Feb 8 19:55:38 2018 by

LaTeXML
structure homomorphism

structure homomorphism

Let

Σ

be a fixed

signature

, and

𝔄

and

𝔅

be two

structures

for

Σ

. The interesting

functions

from

𝔄

to

𝔅

are the ones that

preserve

the structure.

A function

f

:

𝔄

→

𝔅

is said to be a

homomorphism

(or simply

morphism

) if and only if:

1.

For every constant symbol

c

of

Σ

,

f

⁢

(

c

𝔄

)

=

c

𝔅

.

2.

For every

natural number

n

and every

n

-ary function symbol

F

of

Σ

,

f

⁢

(

F

𝔄

⁢

(

a

1

,

…

,

a

n

)

)

=

F

𝔅

⁢

(

f

⁢

(

a

1

)

,

…

,

f

⁢

(

a

n

)

)

.

3.

For every natural number

n

and every

n

-ary relation symbol

R

of

Σ

,

R

𝔄

⁢

(

a

1

,

…

,

a

n

)

⇒

R

𝔅

⁢

(

f

⁢

(

a

1

)

,

…

,

f

⁢

(

a

n

)

)

.

Homomorphisms with various additional

properties

have special names:

•

An

injective

(

http://planetmath.org/Injective

) homomorphism is called a

monomorphism

.

•

A

surjective

homomorphism is called an

epimorphism

.

•

A

bijective

homomorphism is called a

bimorphism

.

•

An injective homomorphism

f

is called an

embedding

if, for every natural number

n

and every

n

-ary relation symbol

R

of

Σ

,

R

𝔅

⁢

(

f

⁢

(

a

1

)

,

…

,

f

⁢

(

a

n

)

)

⇒

R

𝔄

⁢

(

a

1

,

…

,

a

n

)

,

the

converse

of condition 3 above, holds.

•

A surjective embedding is called an

isomorphism

.

•

A homomorphism from a structure to itself (e.g. (

http://planetmath.org/Eg

),

f

:

𝔄

→

𝔄

) is called an

.

•

An isomorphism from a structure to itself is called an

automorphism

.

Title

structure homomorphism

Canonical

name

StructureHomomorphism

Date of creation

2013-03-22 12:43:22

Last modified on

2013-03-22 12:43:22

Owner

almann (2526)

Last modified by

almann (2526)

Numerical id

14

Author

almann (2526)

Entry type

Definition

Classification

msc 03C07

Synonym

homomorphism

Synonym

morphism

Synonym

monomorphism

Synonym

epimorphism

Synonym

bimorphism

Synonym

embedding

Synonym

isomorphism

Synonym

endomorphism

Synonym

automorphism

Related topic

AxiomaticTheoryOfSupercategories

Defines

structure morphism

Defines

structure monomorphism

Defines

structure epimorphism

Defines

structure bimorphism

Defines

structure embedding

Defines

structure isomorphism

Defines

structure endomorphism

Defines

structure automorphism

Generated on Thu Feb 8 19:55:44 2018 by

LaTeXML