aarjaneiro/planetmath huggingface dataset:
stable isomorphism

stable isomorphism

Let

R

be a

ring with unity

1. Two left

R

-modules

M

and

N

are said to be

stably isomorphic

if there exists a

finitely
generated

free

R

-module

R

n

(

n

≥

1

) such that

M

⊕

R

n

≅

N

⊕

R

n

.

A left

R

-module is said to be

stably free

if it is stably isomorphic to a finitely
generated free

R

-module. In other words,

M

is stably free if

M

⊕

R

m

≅

R

n

for some positive integers

m

,

n

.

Remark

In the

Grothendieck group

K

0

⁢

(

R

)

of a ring

R

with 1, two

finitely generated projective module

representatives

M

and

N

such that

[

M

]

=

[

N

]

∈

K

0

⁢

(

R

)

iff they are stably isomorphic
to each other. In particular,

[

M

]

is the

zero element

in

K

0

⁢

(

R

)

iff it is stably free.

Title

stable isomorphism

Canonical name

StableIsomorphism

Date of creation

2013-03-22 15:00:00

Last modified on

2013-03-22 15:00:00

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

4

Author

CWoo (3771)

Entry type

Definition

Classification

msc 19A13

Related topic

AlgebraicKTheory

Defines

stably isomorphic

Defines

stably free

Generated on Fri Feb 9 19:14:36 2018 by

LaTeXML
Steinberg group

Steinberg group

Given an

associative ring

R

with

identity

, the Steinberg group

S

⁢

t

⁢

(

R

)

describes the minimal amount of

relations

between

elementary matrices

in

R

.

For

n

≥

3

, define

S

⁢

t

n

⁢

(

R

)

to be the

free abelian group

on symbols

x

i

⁢

j

⁢

(

r

)

for

i

,

j

distinct integers between

1

and

n

, and

r

∈

R

, subject to the following relations:

x

i

⁢

j

⁢

(

r

)

⁢

x

i

⁢

j

⁢

(

s

)

=

x

i

⁢

j

⁢

(

r

+

s

)

[

x

i

⁢

j

,

x

k

⁢

l

]

=

{

1

if

j

≠

k

and

i

≠

l

x

i

⁢

l

⁢

(

r

⁢

s

)

if

j

=

k

and

i

≠

l

x

k

⁢

j

⁢

(

-

s

⁢

r

)

if

j

≠

k

and

i

=

l

.

Note that if

e

i

⁢

j

⁢

(

r

)

denotes the elementary matrix with one along the

diagonal

, and

r

in the

(

i

,

j

)

entry, then the

e

i

⁢

j

⁢

(

r

)

also satisfy the above relations, giving a well defined morphism

S

⁢

t

n

⁢

(

R

)

→

E

n

⁢

(

R

)

, where the latter is the group of elementary matrices.

Taking a colimit over

n

gives the Steinberg group

S

⁢

t

⁢

(

R

)

. The importance of the Steinberg group is that the kernel of the map

S

⁢

t

⁢

(

R

)

→

E

⁢

(

R

)

is the second algebraic

K

-group of the ring

R

,

K

2

⁢

(

R

)

. This also coincides with the kernel of the Steinberg group. One can also show that the Steinberg group is the universal central extension of the group

E

⁢

(

R

)

.

Title

Steinberg group

Canonical

name

SteinbergGroup

Date of creation

2013-03-22 16:44:38

Last modified on

2013-03-22 16:44:38

Owner

dublisk (96)

Last modified by

dublisk (96)

Numerical id

4

Author

dublisk (96)

Entry type

Definition

Classification

msc 19C09

Generated on Fri Feb 9 19:14:39 2018 by

LaTeXML
Fredholm module

Fredholm module

Fredholm modules

represent abstract elliptic pseudo-differential operators.

Definition 1

.

An

odd Fredholm module

(

ℋ

,

F

)

over a

C

*

-

algebra

A

is given by an involutive

representation

π

of

A

on a

Hilbert space

ℋ

,
together with an

operator

F

on

ℋ

such that

F

=

F

*

,

F

2

=

1

⁢

I

and

[

F

,

π

⁢

(

a

)

]

∈

𝕂

⁢

(

ℋ

)

for all

a

∈

A

.

Definition 2

.

An

even Fredholm module

(

ℋ

,

F

,

Γ

)

is given by
an odd Fredholm module

(

ℋ

,

F

)

together with a

ℤ

2

-grading

Γ

on

ℋ

,

Γ

=

Γ

*

,

Γ

2

=

1

⁢

I

,
such that

Γ

⁢

π

⁢

(

a

)

=

π

⁢

(

a

)

⁢

Γ

and

Γ

⁢

F

=

-

F

⁢

Γ

.

Definition 3

.

A Fredholm module is called

degenerate

if

[

F

,

π

⁢

(

a

)

]

=

0

for all

a

∈

A

.
Degenerate Fredholm modules are

homotopic

to the 0-module.

Example 1

(Fredholm modules over

C

)

An even Fredholm module

(

H

,

F

,

Γ

)

over

C

is given by

ℋ

=

ℂ

k

⊕

ℂ

k

with

⁢

π

⁢

(

a

)

=

(

a

⁢

1

⁢

I

k

0

0

0

)

,

F

=

(

0

1

⁢

I

k

1

⁢

I

k

0

)

,

Γ

=

(

1

⁢

I

k

0

0

-

1

⁢

I

k

)

.

Title

Fredholm module

Canonical name

FredholmModule

Date of creation

2013-03-22 12:57:43

Last modified on

2013-03-22 12:57:43

Owner

mhale (572)

Last modified by

mhale (572)

Numerical id

6

Author

mhale (572)

Entry type

Definition

Classification

msc 19K33

Classification

msc 46L87

Classification

msc 47A53

Related topic

KHomology

Generated on Fri Feb 9 19:14:41 2018 by

LaTeXML
K-homology

K-homology

K-homology is a homology theory on the category of

compact

Hausdorff spaces

.
It classifies the elliptic pseudo-differential operators acting on the
vector bundles over a space.
In terms of

C

*

-algebras, it classifies the

Fredholm modules

over an algebra.

An

operator

homotopy between two Fredholm modules

(

ℋ

,

F

0

,

Γ

)

and

(

ℋ

,

F

1

,

Γ

)

is a norm

continuous

path of Fredholm modules,

t

↦

(

ℋ

,

F

t

,

Γ

)

,

t

∈

[

0

,

1

]

.
Two Fredholm modules are then

equivalent

if they are related by

unitary transformations

or operator homotopies.
The

K

0

⁢

(

A

)

group is the abelian group of

equivalence classes

of even Fredholm modules over A.
The

K

1

⁢

(

A

)

group is the abelian group of equivalence classes
of odd Fredholm modules over A.

Addition

is given by direct summation of Fredholm modules,
and the

inverse

of

(

ℋ

,

F

,

Γ

)

is

(

ℋ

,

-

F

,

-

Γ

)

.

References

1

N. Higson and J. Roe,

Analytic

K-homology

.

Oxford University Press, 2000.

Title

K-homology

Canonical name

Khomology

Date of creation

2013-03-22 12:57:46

Last modified on

2013-03-22 12:57:46

Owner

mhale (572)

Last modified by

mhale (572)

Numerical id

6

Author

mhale (572)

Entry type

Topic

Classification

msc 19K33

Related topic

FredholmModule

Related topic

KTheory

Generated on Fri Feb 9 19:14:43 2018 by

LaTeXML
examples of K-theory groups

examples of K-theory groups

A

K

0

⁢

(

A

)

K

1

⁢

(

A

)

ℂ

ℤ

0

M

n

⁢

(

ℂ

)

ℤ

0

ℍ

ℤ

0

𝕂

ℤ

0

𝔹

0

0

𝔹

/

𝕂

0

ℤ

C

0

⁢

(

(

0

,

1

)

)

0

ℤ

C

0

⁢

(

ℝ

2

⁢

n

)

ℤ

0

C

0

⁢

(

ℝ

2

⁢

n

+

1

)

0

ℤ

C

⁢

(

[

0

,

1

]

)

ℤ

0

C

⁢

(

𝕋

n

)

ℤ

2

n

-

1

ℤ

2

n

-

1

C

⁢

(

𝕊

2

⁢

n

)

ℤ

2

0

C

⁢

(

𝕊

2

⁢

n

+

1

)

ℤ

ℤ

C

⁢

(

ℂ

⁢

P

n

)

ℤ

n

+

1

0

𝒪

n

ℤ

/

(

n

-

1

)

0

A

θ

ℤ

2

ℤ

2

C

*

⁢

(

H

3

)

ℤ

3

ℤ

3

Topological K-theory

of some common

C

*

-algebras.

Title

examples of K-theory groups

Canonical name

ExamplesOfKtheoryGroups

Date of creation

2013-03-22 13:28:44

Last modified on

2013-03-22 13:28:44

Owner

mhale (572)

Last modified by

mhale (572)

Numerical id

8

Author

mhale (572)

Entry type

Example

Classification

msc 19K99

Classification

msc 19L99

Generated on Fri Feb 9 19:14:45 2018 by

LaTeXML
Riemann-Roch theorem for curves

Riemann-Roch theorem for curves

Let

C

be a projective

nonsingular

curve over an algebraically closed field. If

D

is a

divisor

on

C

, then

ℓ

⁢

(

D

)

-

ℓ

⁢

(

K

-

D

)

=

deg

⁢

(

D

)

+

1

-

g

where

g

is the genus of the curve, and

K

is the

canonical

divisor (

ℓ

⁢

(

K

)

=

g

). Here

ℓ

⁢

(

D

)

denotes the

dimension

of the

http://planetmath.org/node/SpaceOfFunctionsAssociatedToADivisor

space of functions associated to a divisor.

Title

Riemann-Roch theorem for curves

Canonical name

RiemannRochTheoremForCurves

Date of creation

2013-03-22 12:03:05

Last modified on

2013-03-22 12:03:05

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

12

Author

mathcam (2727)

Entry type

Theorem

Classification

msc 19L10

Classification

msc 14H99

Related topic

HurwitzGenusFormula

Generated on Fri Feb 9 19:14:47 2018 by

LaTeXML
alternating group has index 2 in the symmetric group, the

alternating group has index 2 in the symmetric group, the

We prove that the

alternating group

A

n

has index 2 in the

symmetric group

S

n

, i.e.,

A

n

has the same

cardinality

as its

complement

S

n

∖

A

n

. The proof is function-theoretic. Its idea is

similar

to the proof in the parent topic, but the focus is less on algebraic aspect.

Let

π

∈

S

n

∖

A

n

. Define

π

:

S

n

∖

A

n

→

A

n

by

π

⁢

(

σ

)

=

π

⁢

σ

, where

π

⁢

σ

is the

product

of

π

and

σ

.

One-to-one

:

π

⁢

(

σ

)

=

π

⁢

(

δ

)

⟹

σ

=

δ

since

π

-

1

exists and

π

-

1

⁢

π

⁢

σ

=

π

-

1

⁢

π

⁢

δ

.

Onto:
Given

α

∈

A

n

, there exists an

element

in

S

n

∖

A

n

, namely

λ

=

π

-

1

⁢

α

, such that

π

⁢

(

α

)

=

λ

.

(The element

λ

is in

S

n

∖

A

n

because

π

-

1

is and the product of an

odd permutation

and an

even permutation

is odd.)

The

function

π

:

S

n

∖

A

n

→

A

n

is, therefore, a

one-to-one correspondence

, so both sets

S

n

∖

A

n

and

A

n

have the same cardinality.

Title

alternating group has index 2 in the symmetric group, the

Canonical name

AlternatingGroupHasIndex2InTheSymmetricGroupThe

Date of creation

2013-03-22 16:48:49

Last modified on

2013-03-22 16:48:49

Owner

yesitis (13730)

Last modified by

yesitis (13730)

Numerical id

8

Author

yesitis (13730)

Entry type

Proof

Classification

msc 20-00

Generated on Fri Feb 9 19:15:01 2018 by

LaTeXML
alternating group is a normal subgroup of the symmetric group

alternating group is a normal subgroup of the symmetric group

Theorem 1

.

The

alternating group

A

n

is a

normal subgroup

of the

symmetric group

S

n

Proof.

Define the

epimorphism

f

:

S

n

→

ℤ

2

by

:

σ

↦

0

if

σ

is an

even permutation

and

:

σ

↦

1

if

σ

is an

odd permutation

. Hence,

A

n

is the kernel of

f

and so it is a normal subgroup of the
domain

S

n

. Furthermore

S

n

/

A

n

≅

ℤ

2

by
the

first isomorphism theorem

. So by Lagrange’s

theorem

|

S

n

|

=

|

A

n

|

⁢

|

S

n

/

A

n

|

.

Therefore,

|

A

n

|

=

n

!

/

2

. That is, there are

n

!

/

2

many

elements

in

A

n

∎

Remark

. What we have shown in the theorem is that, in fact,

A

n

has index

2

in

S

n

. In general, if a

subgroup

H

of

G

has index

2

, then

H

is normal in

G

. (Since

[

G

:

H

]

=

2

, there is an element

g

∈

G

-

H

, so that

g

⁢

H

∩

H

=

∅

and thus

g

⁢

H

=

H

⁢

g

).

Title

alternating group is a normal subgroup of the symmetric group

Canonical name

AlternatingGroupIsANormalSubgroupOfTheSymmetricGroup

Date of creation

2013-03-22 13:42:32

Last modified on

2013-03-22 13:42:32

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

8

Author

CWoo (3771)

Entry type

Theorem

Classification

msc 20-00

Generated on Fri Feb 9 19:15:03 2018 by

LaTeXML
alternative proof of condition on a near ring to be a ring

alternative proof of condition on a near ring to be a ring

Theorem 1

.

Let

(

R

,

+

,

⋅

)

be a near ring with a

multiplicative identity

1

such that the

⋅

also

left distributes over

+

; that is,

c

⋅

(

a

+

b

)

=

c

⋅

a

+

c

⋅

b

. Then

R

is a ring.

Proof.

All that needs to be verified is

commutativity

of

+

.

Let

a

,

b

∈

R

. Consider the expression

(

1

+

1

)

⁢

(

a

+

b

)

.

We have:

(

1

+

1

)

⁢

(

a

+

b

)

=

(

1

+

1

)

⁢

a

+

(

1

+

1

)

⁢

b

by left distributivity

=

1

⁢

a

+

1

⁢

a

+

1

⁢

b

+

1

⁢

b

by right distributivity

=

a

+

a

+

b

+

b

since

⁢

1

⁢

is a multiplicative identity

On the other hand, we have:

(

1

+

1

)

⁢

(

a

+

b

)

=

1

⁢

(

a

+

b

)

+

1

⁢

(

a

+

b

)

by right distributivity

=

a

+

b

+

a

+

b

since

⁢

1

⁢

is a multiplicative identity

Thus,

a

+

a

+

b

+

b

=

a

+

b

+

a

+

b

. Hence:

a

+

b

=

0

+

(

a

+

b

)

+

0

since

⁢

0

⁢

is an additive identity (

http://planetmath.org/AdditiveIdentity

)

=

(

-

a

+

a

)

+

(

a

+

b

)

+

(

b

+

-

b

)

by definition of additive inverse (

http://planetmath.org/AdditiveInverse

)

=

-

a

+

(

a

+

a

+

b

+

b

)

+

-

b

by associativity of

+

=

-

a

+

(

a

+

b

+

a

+

b

)

+

-

b

since

⁢

a

+

a

+

b

+

b

=

a

+

b

+

a

+

b

=

(

-

a

+

a

)

+

(

b

+

a

)

+

(

b

+

-

b

)

by associativity of

+

=

0

+

(

b

+

a

)

+

0

by definition of

=

b

+

a

since

⁢

0

⁢

is an

∎

Title

alternative proof of condition on a near ring to be a ring

Canonical name

AlternativeProofOfConditionOnANearRingToBeARing

Date of creation

2013-03-22 17:20:06

Last modified on

2013-03-22 17:20:06

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

9

Author

Wkbj79 (1863)

Entry type

Proof

Classification

msc 20-00

Classification

msc 16-00

Classification

msc 13-00

Generated on Fri Feb 9 19:15:07 2018 by

LaTeXML
associative

associative

Let

(

S

,

ϕ

)

be a set with

binary operation

ϕ

.

ϕ

is said to be

associative

over

S

if

ϕ

⁢

(

a

,

ϕ

⁢

(

b

,

c

)

)

=

ϕ

⁢

(

ϕ

⁢

(

a

,

b

)

,

c

)

for all

a

,

b

,

c

∈

S

.

Examples of associative operations are

addition

and

multiplication

over the

integers

(or reals), or addition or multiplication over

n

×

n

matrices.

We can construct an operation which is not associative. Let

S

be the integers. and define

ν

⁢

(

a

,

b

)

=

a

2

+

b

. Then

ν

⁢

(

ν

⁢

(

a

,

b

)

,

c

)

=

ν

⁢

(

a

2

+

b

,

c

)

=

a

4

+

2

⁢

b

⁢

a

2

+

b

2

+

c

. But

ν

⁢

(

a

,

ν

⁢

(

b

,

c

)

)

=

ν

⁢

(

a

,

b

2

+

c

)

=

a

+

b

4

+

2

⁢

c

⁢

b

2

+

c

2

, hence

ν

⁢

(

ν

⁢

(

a

,

b

)

,

c

)

≠

ν

⁢

(

a

,

ν

⁢

(

b

,

c

)

)

.

Note, however, that if we were to take

S

=

{

0

}

,

ν

would be associative over

S

!. This illustrates the fact that the set the operation is taken with respect to is very important.

Example.

We show that the

division

operation over nonzero reals is non-associative. All we need is a counter-example: so let us compare

1

/

(

1

/

2

)

and

(

1

/

1

)

/

2

. The first

expression

is equal to

2

, the second to

1

/

2

, hence division over the nonzero reals is not associative.

Remark.

The

property

of being associative of a binary operation can be generalized to an arbitrary

n

-ary operation, where

n

≥

2

. An

n

-ary operation

ϕ

on a set

A

is said to be

associative

if for any elements

a

1

,

…

,

a

2

⁢

n

-

1

∈

A

, we have

ϕ

⁢

(

ϕ

⁢

(

a

1

,

…

,

a

n

)

,

a

n

+

1

⁢

…

,

a

2

⁢

n

-

1

)

=

⋯

=

ϕ

⁢

(

a

1

,

…

,

a

n

-

1

,

ϕ

⁢

(

a

n

,

…

,

a

2

⁢

n

-

1

)

)

.

In other words, for any

i

=

1

,

…

,

n

, if we set

b

i

:=

ϕ

⁢

(

a

1

,

…

,

ϕ

⁢

(

a

i

,

…

,

a

i

+

n

-

1

)

,

…

,

a

2

⁢

n

-

1

)

, then

ϕ

is associative iff

b

i

=

b

1

for all

i

=

1

,

…

,

n

. Therefore, for instance, a ternary operation

f

on

A

is associative if

f

⁢

(

f

⁢

(

a

,

b

,

c

)

,

d

,

e

)

=

f

⁢

(

a

,

f

⁢

(

b

,

c

,

d

)

,

e

)

=

f

⁢

(

a

,

b

,

f

⁢

(

c

,

d

,

e

)

)

.

Title

associative

Canonical

name

Associative

Date of creation

2013-03-22 12:22:48

Last modified on

2013-03-22 12:22:48

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

11

Author

CWoo (3771)

Entry type

Definition

Classification

msc 20-00

Synonym

associativity

Related topic

Commutative

Related topic

Semigroup

Related topic

Group

Defines

non-associative

Generated on Fri Feb 9 19:15:10 2018 by

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Bruhat decomposition

Bruhat decomposition

Bruhat decomposition

is the name for the fact that

B

\

G

/

B

=

W

, where

G

is a reductive group,

B

a Borel subgroup, and

W

the

Weyl group

. Less canonically, one can write

G

=

B

⁢

W

⁢

B

.

In the case of the

general linear group

G

=

G

⁢

L

n

,

B

is the group of nonsingular

upper triangular matrices

, and

W

is the collection of

permutation matrices

(and is

isomorphic

to

S

n

). Any

nonsingular matrix

can thus be written uniquely as a product of an upper triangular matrix, a permutation matrix, and another upper triangular matrix.

Title

Bruhat decomposition

Canonical name

BruhatDecomposition

Date of creation

2013-03-22 15:43:15

Last modified on

2013-03-22 15:43:15

Owner

nerdy2 (62)

Last modified by

nerdy2 (62)

Numerical id

9

Author

nerdy2 (62)

Entry type

Theorem

Classification

msc 20-00

Generated on Fri Feb 9 19:15:12 2018 by

LaTeXML
canonical projection

canonical projection

Given a group

G

and a

normal subgroup

N

⊲

G

there is an

epimorphism

π

:

G

→

G

/

N

defined by sending an element

g

∈

G

to its coset

g

⁢

N

. The epimorphism

π

is referred to as the

canonical projection

.

Title

canonical projection

Canonical name

CanonicalProjection

Date of creation

2013-03-22 13:24:26

Last modified on

2013-03-22 13:24:26

Owner

Dr_Absentius (537)

Last modified by

Dr_Absentius (537)

Numerical id

8

Author

Dr_Absentius (537)

Entry type

Definition

Classification

msc 20-00

Defines

canonical projection

Generated on Fri Feb 9 19:15:14 2018 by

LaTeXML
centralizer

centralizer

Let

G

be a group. The

centralizer

of an

element

a

∈

G

is defined to be the set

C

⁢

(

a

)

=

{

x

∈

G

∣

x

⁢

a

=

a

⁢

x

}

Observe that, by definition,

e

∈

C

⁢

(

a

)

, and that if

x

,

y

∈

C

⁢

(

a

)

, then

x

⁢

y

-

1

⁢

a

=

x

⁢

y

-

1

⁢

a

⁢

(

y

⁢

y

-

1

)

=

x

⁢

y

-

1

⁢

y

⁢

a

⁢

y

-

1

=

x

⁢

a

⁢

y

-

1

=

a

⁢

x

⁢

y

-

1

, so that

x

⁢

y

-

1

∈

C

⁢

(

a

)

. Thus

C

⁢

(

a

)

is a

subgroup

of

G

. For

a

≠

e

, the subgroup is non-trivial, containing at least

{

e

,

a

}

.

To illustrate an application of this

concept

we prove the following lemma.

Lemma:

There exists a

bijection

between the

right cosets

of

C

⁢

(

a

)

and the

conjugates

of

a

.

Proof:

If

x

,

y

∈

G

are in the same right coset, then

y

=

c

⁢

x

for some

c

∈

C

⁢

(

a

)

. Thus

y

-

1

⁢

a

⁢

y

=

x

-

1

⁢

c

-

1

⁢

a

⁢

c

⁢

x

=

x

-

1

⁢

c

-

1

⁢

c

⁢

a

⁢

x

=

x

-

1

⁢

a

⁢

x

.

Conversely

, if

y

-

1

⁢

a

⁢

y

=

x

-

1

⁢

a

⁢

x

then

x

⁢

y

-

1

⁢

a

=

a

⁢

x

⁢

y

-

1

and

x

⁢

y

-

1

∈

C

⁢

(

a

)

giving

x

,

y

are in the same right coset.
Let

[

a

]

denote the

conjugacy class

of

a

. It follows that

|

[

a

]

|

=

[

G

:

C

(

a

)

]

and

|

[

a

]

|

∣

|

G

|

.

We remark that

a

∈

Z

⁢

(

G

)

⇔

C

⁢

(

a

)

=

G

⇔

|

[

a

]

|

=

1

, where

Z

⁢

(

G

)

denotes the center of

G

.

Now let

G

be a

p

-group, i.e. a

finite group

of order

p

n

,
where

p

is a prime and

n

is a

positive integer

.
Let

z

=

|

Z

⁢

(

G

)

|

.

Summing

over elements in distinct conjugacy classes,
we have

p

n

=

∑

|

[

a

]

|

=

z

+

∑

a

∉

Z

⁢

(

G

)

|

[

a

]

|

since the center consists precisely of the conjugacy classes of

cardinality

1

.
But

|

[

a

]

|

∣

p

n

, so

p

∣

z

.
However,

Z

⁢

(

G

)

is certainly non-empty, so we conclude that every

p

-group has a non-trivial center.

The groups

C

⁢

(

g

⁢

a

⁢

g

-

1

)

and

C

⁢

(

a

)

, for any

g

, are

isomorphic

.

Title

centralizer

Canonical name

Centralizer

Date of creation

2013-03-22 12:35:01

Last modified on

2013-03-22 12:35:01

Owner

drini (3)

Last modified by

drini (3)

Numerical id

14

Author

drini (3)

Entry type

Definition

Classification

msc 20-00

Synonym

centraliser

Related topic

Normalizer

Related topic

GroupCentre

Related topic

ClassEquationTheorem

Generated on Fri Feb 9 19:15:16 2018 by

LaTeXML
commutative

commutative

Let

S

be a set and

∘

a

binary operation

on it.

∘

is said to be

commutative

if

a

∘

b

=

b

∘

a

for all

a

,

b

∈

S

.

Viewing

∘

as a

function

from

S

×

S

to

S

, the commutativity of

∘

can be notated as

∘

(

a

,

b

)

=

∘

(

b

,

a

)

.

Some common examples of commutative operations are

•

addition over the integers:

m

+

n

=

m

+

n

for all integers

m

,

n

•

multiplication over the integers:

m

⋅

n

=

m

⋅

n

for all integers

m

,

n

•

addition over

n

×

n

matrices,

A

+

B

=

B

+

A

for all

n

×

n

matrices

A

,

B

, and

•

multiplication over the reals:

r

⁢

s

=

s

⁢

r

, for all real numbers

r

,

s

.

A binary operation that is not commutative is said to be

non-commutative

. A common example of a

non-commutative operation

is the

subtraction

over the integers (or more generally the real numbers). This means that, in general,

a

-

b

≠

b

-

a

.

For instance,

2

-

1

=

1

≠

-

1

=

1

-

2

.

Other examples of non-commutative binary operations can be found in the attachment below.

Remark

. The notion of commutativity can be generalized to

n

-ary operations, where

n

≥

2

. An

n

-ary operation

f

on a set

A

is said to be

commutative

if

f

⁢

(

a

1

,

a

2

,

…

,

a

n

)

=

f

⁢

(

a

π

⁢

(

1

)

,

a

π

⁢

(

2

)

,

…

,

a

π

⁢

(

n

)

)

for every

permutation

π

on

{

1

,

2

,

…

,

n

}

, and for every choice of

n

elements

a

i

of

A

.

Title

commutative

Canonical name

Commutative

Date of creation

2013-03-22 12:22:45

Last modified on

2013-03-22 12:22:45

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

11

Author

CWoo (3771)

Entry type

Definition

Classification

msc 20-00

Synonym

commutativity

Synonym

commutative law

Related topic

Associative

Related topic

AbelianGroup2

Related topic

QuantumTopos

Related topic

NonCommutativeStructureAndOperation

Related topic

Subcommutative

Defines

non-commutative

Generated on Fri Feb 9 19:15:18 2018 by

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