aarjaneiro/planetmath huggingface dataset:
abelian group

abelian group

Let

(

G

,

*

)

be a group. If for any

a

,

b

∈

G

we have

a

*

b

=

b

*

a

, we say that the group is

abelian

(or

commutative

).

Abelian groups

are named after Niels Henrik Abel, but the word

abelian

is commonly written in lowercase.

Abelian groups are essentially the same thing as

unitary

ℤ

-modules (

http://planetmath.org/Module

).
In fact, it is often more natural to treat abelian groups as modules rather than as groups, and for this reason they are commonly written using

additive

notation.

Some of the basic properties of abelian groups are as follows:

Theorem 1

.

Any

subgroup

(

http://planetmath.org/Subgroup

) of an abelian group is normal.

Proof.

Let

H

be a subgroup of the abelian group

G

. Since

a

⁢

h

=

h

⁢

a

for any

a

∈

G

and any

h

∈

H

we get

a

⁢

H

=

H

⁢

a

. That is,

H

is normal in

G

.
∎

Theorem 2

.

Quotient groups

of abelian groups are also abelian.

Proof.

Let

H

be a subgroup of

G

. Since

G

is abelian,

H

is normal and we can get the quotient group

G

/

H

whose

elements

are the

equivalence classes

for

a

∼

b

if

a

⁢

b

-

1

∈

H

.
The

operation

on the quotient group is given by

a

⁢

H

⋅

b

⁢

H

=

(

a

⁢

b

)

⁢

H

. But

b

⁢

H

⋅

a

⁢

H

=

(

b

⁢

a

)

⁢

H

=

(

a

⁢

b

)

⁢

H

, therefore the quotient group is also commutative.
∎

Here is another

theorem

concerning abelian groups:

Theorem 3

.

If

φ

:

G

→

G

defined by

φ

⁢

(

x

)

=

x

2

is a

homomorphism

(

http://planetmath.org/GroupHomomorphism

), then

G

is abelian.

Proof.

If such a

function

were a homomorphism,
we would have

(

x

⁢

y

)

2

=

φ

⁢

(

x

⁢

y

)

=

φ

⁢

(

x

)

⁢

φ

⁢

(

y

)

=

x

2

⁢

y

2

that is,

x

⁢

y

⁢

x

⁢

y

=

x

⁢

x

⁢

y

⁢

y

.
Left-multiplying by

x

-

1

and right-multiplying by

y

-

1

we are led to

y

⁢

x

=

x

⁢

y

and thus the group is abelian.
∎

Title

abelian group

Canonical name

AbelianGroup

Date of creation

2013-03-22 14:01:55

Last modified on

2013-03-22 14:01:55

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

25

Author

yark (2760)

Entry type

Definition

Classification

msc 20K99

Synonym

commutative group

Related topic

Group

Related topic

Klein4Group

Related topic

CommutativeSemigroup

Related topic

GeneralizedCyclicGroup

Related topic

AbelianGroupsOfOrder120

Related topic

FundamentalTheoremOfFinitelyGeneratedAbelianGroups

Related topic

NonabelianGroup

Related topic

Commutative

Related topic

MetabelianGroup

Defines

abelian

Defines

commutative

Generated on Fri Feb 9 19:32:24 2018 by

LaTeXML
abelian group is divisible if and only if it is an injective object

abelian group is divisible if and only if it is an injective object

Proposition

.

Abelian group

A

is divisible if and only if

A

is an

injective object

in the category of abelian groups.

Proof.

,,

⇐

” Assume that

A

is not divisible, i.e. there exists

a

∈

A

and

n

∈

ℕ

such that the

equation

n

⁢

x

=

a

has no solution in

A

. Let

B

=

<

a

>

be a

cyclic subgroup

generated by

a

and

i

:

B

→

A

the

canonical

inclusion

. Now there are two possibilities: either

B

is finite or

infinite

.

If

B

is infinite, then let

H

=

ℤ

and let

f

:

B

→

H

be defined on

generator

by

f

⁢

(

a

)

=

n

. Now

A

is

injective

, thus there exists

h

:

H

→

A

such that

h

∘

f

=

i

. Thus

n

⋅

h

⁢

(

1

)

=

h

⁢

(

1

)

+

⋯

+

h

⁢

(

1

)

=

h

⁢

(

1

+

⋯

+

1

)

=

h

⁢

(

n

)

=

h

⁢

(

f

⁢

(

a

)

)

=

i

⁢

(

a

)

=

a

.

Contradiction

with

definition

of

n

∈

ℕ

and

a

∈

A

.

If

B

is finite, then let

k

=

|

B

|

(note that

n

does not divide

k

) and let

H

=

ℤ

n

⋅

k

. Furtheremore define

f

:

B

→

H

on generator by

f

⁢

(

a

)

=

n

(note that in this case

f

is a well defined

homomorphism

). Again injectivity of

A

implies

existence

of

h

:

H

→

A

such that

h

∘

f

=

i

. Similarly we get contradiction:

n

⋅

h

⁢

(

1

)

=

h

⁢

(

1

)

+

⋯

+

h

⁢

(

1

)

=

h

⁢

(

1

+

⋯

+

1

)

=

h

⁢

(

n

)

=

h

⁢

(

f

⁢

(

a

)

)

=

i

⁢

(

a

)

=

a

.

This

completes

first

implication

.

,,

⇒

” This implication is proven here (

http://planetmath.org/ExampleOfInjectiveModule

).

□

Remark.

It is clear that in the category of abelian groups

𝒜

⁢

ℬ

, a group

A

is projective if and only if

A

is free. This is since

𝒜

⁢

ℬ

is

equivalent

to the

category

of

ℤ

-modules and

projective modules

are

direct summands

of

free modules

. Since

ℤ

is a

principal ideal domain

, then every

submodule

of a free module is free, thus projective

ℤ

-modules are free.

Title

abelian group is divisible if and only if it is an injective object

Canonical name

AbelianGroupIsDivisibleIfAndOnlyIfItIsAnInjectiveObject

Date of creation

2013-03-22 18:48:15

Last modified on

2013-03-22 18:48:15

Owner

joking (16130)

Last modified by

joking (16130)

Numerical id

8

Author

joking (16130)

Entry type

Theorem

Classification

msc 20K99

Generated on Fri Feb 9 19:32:26 2018 by

LaTeXML
difference

difference

The

difference

of

two numbers

a

and

b

is a number

d

such that

b

+

d

=

a

.

The difference of

a

(the

minuend

) and

b

(the

subtrahend

) is denoted by

a

-

b

.

The definition is for the elements

a

,

b

of any

Abelian group

(e.g. of a

vector space

). The difference of them is always unique.

Note 1.

Forming the difference of numbers (resp. elements), i.e.

subtraction

, is in a certain sense converse to the addition operation:

(

x

+

y

)

-

y

=

x

Note 2.

As for

real numbers

, one may say that the difference

between

a

and

b

is

|

a

-

b

|

(which is the same as

|

b

-

a

|

); then it is always nonnegative.  For all

complex numbers

, such a phrase would be nonsense.

Some

•

b

+

(

a

-

b

)

=

a

•

a

-

b

=

a

+

(

-

b

)

•

-

(

a

-

b

)

=

b

-

a

•

n

(

a

-

b

)

=

n

a

-

n

b

(

n

∈

ℤ

)

•

a

-

a

=

0

Title

difference

Canonical name

Difference

Date of creation

2013-03-22 17:33:35

Last modified on

2013-03-22 17:33:35

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

16

Author

pahio (2872)

Entry type

Definition

Classification

msc 20K99

Classification

msc 00A05

Classification

msc 11B25

Related topic

VectorDifference

Related topic

SetDifference

Related topic

Multiple

Related topic

GeneralAssociativity

Related topic

Quotient

Related topic

DifferenceOfVectors

Defines

minuend

Defines

subtrahend

Generated on Fri Feb 9 19:32:28 2018 by

LaTeXML
divisible group

divisible group

An

abelian group

D

is said to be divisible if for any

x

∈

D

,

n

∈

ℤ

+

, there exists an

element

x

′

∈

D

such that

n

⁢

x

′

=

x

.

Some noteworthy facts:

•

An abelian group is

injective

(

http://planetmath.org/InjectiveModule

) (as a

ℤ

-module) if and only if it is divisible.

•

Every group is

isomorphic

to a

subgroup

of a

divisible group

.

•

Any divisible abelian group is isomorphic to the

direct sum

of its

torsion subgroup

and

n

copies of the group of

rationals

(for some

cardinal number

n

).

Title

divisible group

Canonical name

DivisibleGroup

Date of creation

2013-03-22 13:47:17

Last modified on

2013-03-22 13:47:17

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

7

Author

mathcam (2727)

Entry type

Definition

Classification

msc 20K99

Generated on Fri Feb 9 19:32:30 2018 by

LaTeXML
example of divisible group

example of divisible group

Let

G

denote the group of

rational numbers

taking the

operation

to be

addition

. Then for any

p

q

∈

G

and

n

∈

ℤ

+

, we have

p

n

⁢

q

∈

G

satisfying

n

⁢

p

n

⁢

q

=

p

q

, so the group is

divisible

.

Title

example of divisible group

Canonical name

ExampleOfDivisibleGroup

Date of creation

2013-03-22 13:47:20

Last modified on

2013-03-22 13:47:20

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

4

Author

mathcam (2727)

Entry type

Example

Classification

msc 20K99

Generated on Fri Feb 9 19:32:32 2018 by

LaTeXML
groups in field

groups in field

If

(

K

,

+

,

⋅

)

is a field, then

•

(

K

,

+

)

is the

additive group

of the field

,

•

(

K

∖

{

0

}

,

⋅

)

is the

multiplicative group

of the field

.

Both of these groups are

Abelian

.

The former has always as a

subgroup

{

n

⋅

1

⁢

⋮

⁢

n

∈

ℤ

}

,

the group of the

multiples

of unity.  This is, apparently,

isomorphic

to
the additive group

ℤ

or

ℤ

p

depending on whether the

characteristic

(

http://planetmath.org/Characteristic

) of the field is 0 or a

prime number

p

.

The multiplicative group of any field has as its subgroup the set

E

consisting of all

roots of unity

in the field.  The group

E

has the subgroup

{

1

,

-

1

}

which reduces to

{

1

}

if the of the field is two.

Example 1.

The additive group

(

ℝ

,

+

)

of the reals is isomorphic to the multiplicative group

(

ℝ

+

,

⋅

)

of the

positive

reals; the isomorphy is implemented e.g. by the

isomorphism

mapping

x

↦

2

x

.

Example 2.

Suppose that the of

K

is not 2 and denote the multiplicative group of

K

by

K

*

.  We can consider the four functions

f

i

:

K

*

→

K

*

defined by

f

0

⁢

(

x

)

:=

x

,

f

1

⁢

(

x

)

:=

-

x

,

f

2

⁢

(

x

)

:=

x

-

1

,

f

3

⁢

(

x

)

:=

-

x

-

1

.  The composition of functions is a

binary operation

of the set

G

=

{

f

0

,

f

1

,

f

2

,

f

3

}

,  and we see that

G

is isomorphic to Klein’s 4-group.

Note.

One may also speak of the

additive group

of any

ring

.  Every ring contains also its group of units.

Title

groups in field

Canonical name

GroupsInField

Date of creation

2013-03-22 14:41:58

Last modified on

2013-03-22 14:41:58

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

24

Author

pahio (2872)

Entry type

Topic

Classification

msc 20K99

Classification

msc 20F99

Classification

msc 20A05

Classification

msc 12E99

Related topic

Klein4Group

Related topic

Klein4Ring

Related topic

GroupsOfRealNumbers

Defines

additive group of the field

Defines

multiplicative group of the field

Defines

additive group

Defines

multiplicative group

Generated on Fri Feb 9 19:32:34 2018 by

LaTeXML
locally cyclic group

locally cyclic group

Definition

A

locally cyclic

group is a group in which every finite subset

generates

a

cyclic subgroup

.

Properties

From the definition we see that every

finitely generated

locally cyclic group
(and, in particular, every finite locally cyclic group) is cyclic.

The following can all be shown to be

equivalent

for a group

G

:

•

G

is locally cyclic.

•

For all

a

,

b

∈

G

, the

subgroup

(

http://planetmath.org/Subgroup

)

⟨

a

,

b

⟩

is cyclic.

•

G

is the union of a chain of cyclic subgroups.

•

The

lattice of subgroups

of

G

is

distributive

(

http://planetmath.org/DistributiveLattice

).

•

G

embeds in

ℚ

or

ℚ

/

ℤ

.

•

G

is

isomorphic

to a subgroup of a

quotient

(

http://planetmath.org/QuotientGroup

) of

ℚ

.

•

G

is

involved in

(

http://planetmath.org/SectionOfAGroup

)

ℚ

.

From the last of these equivalent properties it is clear that
every locally cyclic group is

countable

and

abelian

,
and that subgroups and quotients
of locally cyclic groups are locally cyclic.

Title

locally cyclic group

Canonical name

LocallyCyclicGroup

Date of creation

2013-03-22 13:47:12

Last modified on

2013-03-22 13:47:12

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

24

Author

yark (2760)

Entry type

Definition

Classification

msc 20K99

Classification

msc 20E25

Synonym

locally cyclic

Synonym

generalized cyclic group

Synonym

generalized cyclic

Synonym

generalised cyclic

Synonym

generalised cyclic group

Related topic

CyclicGroup

Related topic

AbelianGroup2

Related topic

Group

Related topic

LocallyCalP

Generated on Fri Feb 9 19:32:41 2018 by

LaTeXML
n-divisible group

n

-divisible group

Let

n

be a

positive integer

and

G

an

abelian group

. An

element

x

∈

G

is said to be

divisible

by

n

if there is

y

∈

G

such that

x

=

n

⁢

y

.

By the unique factorization of

ℤ

, write

n

=

p

1

m

1

⁢

p

2

m

2

⁢

⋯

⁢

p

k

m

k

where each

p

i

is a

prime number

(distinct from one another) and

m

i

a positive integer.

Proposition 1

.

If

x

is divisible by

n

, then

x

is divisible by

p

1

,

p

2

,

…

,

p

k

.

Proof.

If

x

is divisible by

n

, write

x

=

n

⁢

y

, where

y

∈

G

. Since

p

i

divides

n

, write

n

=

p

i

⁢

t

i

where

t

i

is a positive integer. Then

x

=

p

i

⁢

t

i

⁢

(

y

)

=

p

i

⁢

(

t

i

⁢

y

)

. Since

t

i

⁢

y

∈

G

,

x

is divisible by

p

i

.
∎

Definition

. An abelian group

G

such that every element is divisible by

n

is called an

n

-

divisible group

. Clearly, every group is

1

-divisible.

For example, the subset

D

⊆

ℚ

of all

decimal fractions

is

10

-divisible.

D

is also

2

and

5

-divisible. In general, we have the following:

Proposition 2

.

If

G

is

n

-divisible, it is also

n

s

-divisible for every non-negative

integer

s

.

Proposition 3

.

Suppose

p

and

q

are

coprime

, then

G

is

p

-divisible and

q

-divisible iff it is

p

⁢

q

-divisible.

Proof.

This follows from

proposition

1 and the fact that if

p

|

n

,

q

|

n

and

gcd

⁡

(

p

,

q

)

=

1

, then

p

q

|

n

. ∎

Proposition 4

.

G

is

n

-divisible iff

G

is

p

-divisible for every prime

p

dividing

n

.

Proof.

Suppose

G

is

n

-divisible. By proposition 1, every element

x

∈

G

is divisible by

p

, so that

G

is

p

-divisible.

Conversely

, suppose

G

is

p

-divisible for every

p

|

n

. Write

n

=

p

1

m

1

⁢

p

2

m

2

⁢

⋯

⁢

p

k

m

k

. Then if

G

is

p

i

m

i

-divisible for every

i

=

1

,

…

,

k

. Since

p

i

m

i

and

p

j

m

j

are coprime,

G

is

n

-divisible by

induction

and proposition 3.
∎

Remark

.

G

is a divisible group iff

G

is

p

-divisible for every prime

p

.

Title

n

-divisible group

Canonical

name

NdivisibleGroup

Date of creation

2013-03-22 17:27:30

Last modified on

2013-03-22 17:27:30

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

5

Author

CWoo (3771)

Entry type

Definition

Classification

msc 20K99

Synonym

n-divisible group

Defines

n-divisible

Defines

n

-divisible

Generated on Fri Feb 9 19:32:53 2018 by

LaTeXML
example of Munn tree

example of Munn tree

Let

X

=

{

a

,

b

,

c

}

,

w

=

a

⁢

a

⁢

a

-

1

⁢

a

-

1

⁢

a

-

1

⁢

a

⁢

b

⁢

b

-

1

⁢

a

⁢

b

-

1

⁢

b

⁢

c

⁢

a

⁢

a

-

1

⁢

c

⁢

c

-

1

. The

reduced

prefix set

of

w

is

red

⁢

(

pref

⁢

(

w

)

)

=

{

ε

,

a

,

a

⁢

a

,

a

-

1

,

b

,

a

⁢

b

-

1

,

a

⁢

c

,

a

⁢

c

⁢

a

,

a

⁢

c

⁢

c

}

.

The

Munn tree

MT

⁢

(

w

)

is the following.

\xymatrix

⁢

&

⁢

b

⁢

&

⁢

a

⁢

b

-

1

⁢

\ar

⁢

[

d

]

b

⁢

&

⁢

a

-

1

⁢

\ar

⁢

[

r

]

a

⁢

&

⁢

ε

⁢

\ar

⁢

[

u

]

b

⁢

\ar

⁢

[

r

]

a

⁢

&

⁢

a

⁢

\ar

⁢

[

d

]

c

⁢

\ar

⁢

[

r

]

a

⁢

&

⁢

a

⁢

a

⁢

&

⁢

&

⁢

a

⁢

c

⁢

\ar

⁢

[

r

]

c

⁢

\ar

⁢

[

d

]

a

⁢

&

⁢

a

⁢

c

⁢

c

⁢

&

⁢

&

⁢

a

⁢

c

⁢

a

⁢

&

Note that we have drawn only edges of the form

(

v

1

,

x

,

v

2

)

(i.e.

\xymatrix

⁢

v

1

⁢

\ar

⁢

[

r

]

x

⁢

&

⁢

v

2

) with

x

∈

X

, leaving implicit the existence of the opposite edges

(

v

2

,

x

-

1

,

v

1

)

(i.e.

\xymatrix

⁢

v

2

⁢

\ar

⁢

[

r

]

x

-

1

⁢

&

⁢

v

1

), as usual in the diagram

representation

of

inverse

word graphs.

Title

example of Munn tree

Canonical name

ExampleOfMunnTree

Date of creation

2013-03-22 16:12:02

Last modified on

2013-03-22 16:12:02

Owner

Mazzu (14365)

Last modified by

Mazzu (14365)

Numerical id

20

Author

Mazzu (14365)

Entry type

Example

Classification

msc 20M05

Classification

msc 20M18

Generated on Fri Feb 9 19:33:21 2018 by

LaTeXML
free hull

free hull

Let

A

be an arbitrary set,
let

A

∗

be the

free monoid

on

A

,
and let

X

be a subset of

A

∗

.

It follows from the

characterization of free submonoids

that the

intersection

M

of all the

free submonoids

of

A

∗

that contain

X

is a free submonoid of

A

∗

.
The

minimal

generating set

H

of

M

is called the

free hull

of

X

.

Title

free hull

Canonical name

FreeHull

Date of creation

2013-03-22 18:21:40

Last modified on

2013-03-22 18:21:40

Owner

Ziosilvio (18733)

Last modified by

Ziosilvio (18733)

Numerical id

4

Author

Ziosilvio (18733)

Entry type

Definition

Classification

msc 20M05

Classification

msc 20M10

Generated on Fri Feb 9 19:33:23 2018 by

LaTeXML
prefix set

prefix set

Let

X

be a set, and

w

∈

X

*

be a word, i.e. an element of the

free monoid

on

X

. A word

v

∈

X

*

is called

prefix

of

w

when a second word

z

∈

X

*

exists such that

x

=

v

⁢

z

. A

proper prefix

of a word u is a prefix v of u not equal to u (sometimes v is required to be non-empty).

Note that the

empty word

ε

and

w

are prefix of

w

, and a proper prefix of

w

if

w

is non-empty.

The

prefix set

of

w

is the set

pref

⁢

(

w

)

of prefixes of

w

, i.e. if

w

=

w

1

⁢

w

2

⁢

…

⁢

w

n

with

w

j

∈

X

for each

j

∈

{

1

,

…

,

n

}

we have

pref

⁢

(

w

)

=

{

ε

,

w

1

,

w

1

⁢

w

2

,

…

,

w

1

⁢

w

2

⁢

…

⁢

w

n

-

1

,

w

}

.

Some closely related concepts are:

1.

A set of words is

prefix closed

if for every word in the set, any of its prefix is also in the set.

2.

The

prefix closure

of a set S is the smallest prefix closed set containing S, or, equivalently, the union of the prefix sets of words in S.

3.

A set S is

prefix free

if for any word in S, no proper prefixes of the word are in S.

Title

prefix set

Canonical

name

PrefixSet

Date of creation

2013-03-22 16:11:56

Last modified on

2013-03-22 16:11:56

Owner

Mazzu (14365)

Last modified by

Mazzu (14365)

Numerical id

6

Author

Mazzu (14365)

Entry type

Definition

Classification

msc 20M05

Defines

prefix

Defines

prefix set

Defines

proper prefix

Defines

prefix closed

Defines

prefix closure

Defines

prefix free

Generated on Fri Feb 9 19:33:29 2018 by

LaTeXML
presentation of inverse monoids and inverse semigroups

presentation of inverse monoids and inverse semigroups

Let

(

X

∐

X

-

1

)

∗

be the

free monoid
with involution

on

X

, and

T

⊆

(

X

∐

X

-

1

)

∗

×

(

X

∐

X

-

1

)

∗

be a

binary relation

between words. We denote by

T

e

[resp.

T

c

] the

equivalence relation

[resp.

congruence

] generated by

T

.

A

presentation

(for an

inverse

monoid)

is a couple

(

X

;

T

)

. We use this couple of objects to define an inverse monoid

Inv

1

⁢

⟨

X

|

T

⟩

. Let

ρ

X

be the

Wagner congruence

on

X

, we define the inverse monoid

Inv

1

⁢

⟨

X

|

T

⟩

presented

by

(

X

;

T

)

as

Inv

1

⁢

⟨

X

|

T

⟩

=

(

X

∐

X

-

1

)

∗

/

(

T

∪

ρ

X

)

c

.

In the previous dicussion, if we replace everywhere

(

X

∐

X

-

1

)

∗

with

(

X

∐

X

-

1

)

+

we obtain a

presentation (for an inverse semigroup)

(

X

;

T

)

and an inverse semigroup

Inv

⁢

⟨

X

|

T

⟩

presented

by

(

X

;

T

)

.

A

trivial

but important example is the Free Inverse Monoid [resp. Free Inverse Semigroup] on

X

, that is usually denoted by

FIM

⁢

(

X

)

[resp.

FIS

⁢

(

X

)

] and is defined by

FIM

⁢

(

X

)

=

Inv

1

⁢

⟨

X

|

∅

⟩

=

(

X

∐

X

-

1

)

∗

/

ρ

X

,

[resp.

FIS

⁢

(

X

)

=

Inv

⁢

⟨

X

|

∅

⟩

=

(

X

∐

X

-

1

)

+

/

ρ

X

]

.

References

1

N. Petrich,

Inverse Semigroups

, Wiley, New York, 1984.

2

J.B. Stephen,

Presentation of inverse monoids

, J. Pure Appl.

Algebra

63 (1990) 81-112.

Title

presentation of inverse monoids and inverse semigroups

Canonical

name

PresentationOfInverseMonoidsAndInverseSemigroups

Date of creation

2013-03-22 16:11:01

Last modified on

2013-03-22 16:11:01

Owner

Mazzu (14365)

Last modified by

Mazzu (14365)

Numerical id

10

Author

Mazzu (14365)

Entry type

Definition

Classification

msc 20M05

Classification

msc 20M18

Synonym

presentation

Synonym

generators and relators

Generated on Fri Feb 9 19:33:33 2018 by

LaTeXML
Schützenberger graph

Schützenberger graph

Let

(

X

;

T

)

be a

presentation

for the

inverse

monoid

Inv

1

⁢

⟨

X

|

T

⟩

[resp. inverse semigroup

Inv

⁢

⟨

X

|

T

⟩

]. In what follows, the

argument

for inverse semigroups and inverse monoids is exactly the same, so we concentrate on the last one.

Given

m

∈

Inv

1

⁢

⟨

X

|

T

⟩

, let

[

m

]

ℛ

be the

equivalence class

of

m

with respect to the Right

Green relation

ℛ

. The

Right Schützenberger graph

of

[

m

]

ℛ

with respect to the presentation

(

X

;

T

)

is defined as the

X

-inverse word graph

𝒮

⁢

Γ

⁢

(

X

;

T

;

m

)

with vertex and

edge set

respectively

V

⁢

(

𝒮

⁢

Γ

⁢

(

X

;

T

;

m

)

)

=

{

v

∈

Inv

1

⁢

⟨

X

|

T

⟩

|

[

v

]

ℛ

=

[

m

]

ℛ

}

,

E

⁢

(

𝒮

⁢

Γ

⁢

(

X

;

T

;

m

)

)

=

{

(

v

1

,

x

,

v

2

)

|

v

1

,

v

2

∈

V

⁢

(

𝒮

⁢

Γ

⁢

(

X

;

T

;

m

)

)

,

x

∈

(

X

∐

X

-

1

)

,

v

2

=

v

1

⋅

[

x

]

τ

}

,

where

τ

=

(

T

∪

ρ

X

)

c

, i.e.

τ

is the

congruence generated by

T

and the

Wagner congruence

ρ

X

, and

[

x

]

τ

is the congruence class of the letter

x

∈

(

X

∐

X

-

1

)

with respect to the

congruence

τ

.

This is a good

definition

, in fact it can be easily shown that given

m

,

n

∈

Inv

1

⁢

⟨

X

|

T

⟩

with

[

m

]

ℛ

=

[

n

]

ℛ

we have

𝒮

⁢

Γ

⁢

(

X

;

T

;

m

)

=

𝒮

⁢

Γ

⁢

(

X

;

T

;

n

)

.

Analogously we can define the

Left Schützenberger graph

using the Left Green relation

ℒ

instead of the Right Green relation

ℛ

, but this notion is not used in literature.

Schützenberger graphs play in combinatorial inverse semigroups theory the role that

Cayley graphs

play in combinatorial group theory. In fact, if

G

=

Inv

1

⁢

⟨

X

|

T

⟩

happen to be a group (with

identity

1

G

), then the Schützenberger graph

𝒮

⁢

Γ

⁢

(

X

;

T

;

1

G

)

of its unique

ℛ

-class is exactly the Cayley graph of the group

G

.

References

1

N. Petrich,

Inverse Semigroups

, Wiley, New York, 1984.

2

J.B. Stephen,

Presentation of inverse monoids

, J. Pure Appl.

Algebra

63 (1990) 81-112.

Title

Schützenberger graph

Canonical

name

SchutzenbergerGraph

Date of creation

2013-03-22 16:10:50

Last modified on

2013-03-22 16:10:50

Owner

Mazzu (14365)

Last modified by

Mazzu (14365)

Numerical id

34

Author

Mazzu (14365)

Entry type

Definition

Classification

msc 20M05

Classification

msc 20M18

Related topic

MunnTree

Defines

Schützenberger graph

Defines

left Schützenberger graph

Defines

right Schützenberger graph

Generated on Fri Feb 9 19:33:36 2018 by

LaTeXML