aarjaneiro/planetmath huggingface dataset:
orbit-stabilizer theorem

orbit-stabilizer theorem

Suppose that

G

is a group acting (

http://planetmath.org/GroupAction

) on a set

X

.
For each

x

∈

X

, let

G

⁢

x

be the orbit of

x

,
let

G

x

be the

stabilizer

of

x

,
and let

ℒ

x

be the set of

left cosets

of

G

x

.
Then for each

x

∈

X

the

function

f

:

G

⁢

x

→

ℒ

x

defined by

g

⁢

x

↦

g

⁢

G

x

is a

bijection

.
In particular,

|

G

x

|

=

[

G

:

G

x

]

and

|

G

⁢

x

|

⋅

|

G

x

|

=

|

G

|

for all

x

∈

X

.

Proof

:

If

y

∈

G

⁢

x

is such that

y

=

g

1

⁢

x

=

g

2

⁢

x

for some

g

1

,

g

2

∈

G

,
then we have

g

2

-

1

⁢

g

1

⁢

x

=

g

2

-

1

⁢

g

2

⁢

x

=

1

⁢

x

=

x

, and so

g

2

-

1

⁢

g

1

∈

G

x

,
and therefore

g

1

⁢

G

x

=

g

2

⁢

G

x

.
This shows that

f

is

well-defined

.

It is clear that

f

is

surjective

.
If

g

⁢

G

x

=

g

′

⁢

G

x

, then

g

=

g

′

⁢

h

for some

h

∈

G

x

,
and so

g

⁢

x

=

(

g

′

⁢

h

)

⁢

x

=

g

′

⁢

(

h

⁢

x

)

=

g

′

⁢

x

.
Thus

f

is also

injective

.

Title

orbit-stabilizer theorem

Canonical

name

OrbitstabilizerTheorem

Date of creation

2013-03-22 12:23:10

Last modified on

2013-03-22 12:23:10

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

22

Author

yark (2760)

Entry type

Theorem

Classification

msc 20M30

Generated on Fri Feb 9 19:34:37 2018 by

LaTeXML
proof of counting theorem

proof of counting theorem

Let

N

be the

cardinality

of the set of all the couples

(

g

,

x

)

such that

g

⋅

x

=

x

. For each

g

∈

G

, there exist

stab

g

⁡

(

X

)

couples with

g

as the first

element

, while for each

x

, there are

|

G

x

|

couples with

x

as the second element. Hence the following equality holds:

N

=

∑

g

∈

G

stab

g

⁡

(

X

)

=

∑

x

∈

X

|

G

x

|

.

From the

orbit-stabilizer theorem

it follows that:

N

=

|

G

|

⁢

∑

x

∈

X

1

|

G

⁢

(

x

)

|

.

Since all the

x

belonging to the same orbit

G

⁢

(

x

)

contribute with

|

G

⁢

(

x

)

|

⁢

1

|

G

⁢

(

x

)

|

=

1

in the sum, then

∑

x

∈

X

1

/

|

G

⁢

(

x

)

|

precisely equals the number of distinct orbits

s

. We have therefore

∑

g

∈

G

stab

g

⁡

(

X

)

=

|

G

|

⁢

s

,

which proves the

theorem

.

Title

proof of counting theorem

Canonical name

ProofOfCountingTheorem

Date of creation

2013-03-22 12:47:07

Last modified on

2013-03-22 12:47:07

Owner

n3o (216)

Last modified by

n3o (216)

Numerical id

5

Author

n3o (216)

Entry type

Proof

Classification

msc 20M30

Generated on Fri Feb 9 19:34:39 2018 by

LaTeXML
simply transitive

simply transitive

Let

G

be a group acting on a set

X

. The action is said to be

simply

transitive

if it is transitive and

∀

x

,

y

∈

X

there is a

unique

g

∈

G

such that

g

.

x

=

y

.

Theorem

.

A

group action

is simply transitive if and only if it is free and transitive

Proof.

Necessity

follows since

g

.

x

=

x

implies that

g

=

1

G

because

1

G

.

x

=

x

also. Now assume the action is free and transitive and we have

elements

g

1

,

g

2

∈

G

and

x

,

y

∈

X

such that

g

1

.

x

=

y

and

g

2

.

x

=

y

. Then

g

1

.

x

=

g

2

.

x

⟹

g

2

-

1

.

g

1

.

x

=

(

g

2

-

1

⁢

g

1

)

.

x

=

x

hence

g

2

-

1

⁢

g

1

=

1

G

because the action is free. Thus

g

1

=

g

2

and so the action is simply transitive.
∎

Title

simply transitive

Canonical name

SimplyTransitive

Date of creation

2013-03-22 14:37:41

Last modified on

2013-03-22 14:37:41

Owner

benjaminfjones (879)

Last modified by

benjaminfjones (879)

Numerical id

7

Author

benjaminfjones (879)

Entry type

Definition

Classification

msc 20M30

Related topic

GroupAction

Generated on Fri Feb 9 19:34:41 2018 by

LaTeXML
stabilizer

stabilizer

Let

G

be a group,

X

a set, and

⋅

:

G

×

X

⟶

X

a

group action

. For any subset

S

of

X

, the

stabilizer

of

S

, denoted

Stab

⁡

(

S

)

, is the

subgroup

Stab

⁡

(

S

)

:=

{

g

∈

G

∣

g

⋅

s

∈

S

⁢

for all

⁢

s

∈

S

}

.

The stabilizer of a single point

x

in

X

is often denoted

G

x

.

Title

stabilizer

Canonical name

Stabilizer

Date of creation

2013-03-22 12:12:22

Last modified on

2013-03-22 12:12:22

Owner

djao (24)

Last modified by

djao (24)

Numerical id

7

Author

djao (24)

Entry type

Definition

Classification

msc 20M30

Classification

msc 16W22

Synonym

isotropy subgroup

Generated on Fri Feb 9 19:34:42 2018 by

LaTeXML
automaton over a monoid

automaton over a monoid

Recall that a

semiautomaton

A

can be visually represented by a

directed graph

G

A

, whose vertices (or nodes) are states of

A

, and whose edges are labeled by

elements

from the input

alphabet

Σ

of

A

.

Labeling

can be extended to paths of

G

A

in a natural way: if

(

e

1

,

…

,

e

n

)

is a path

p

and each edge

e

i

is labeled

a

i

, then the label of

p

is

a

1

⁢

⋯

⁢

a

n

. Thus, the labels of paths of

G

A

are just elements of the monoid

Σ

*

. The

concept

can be readily generalized to arbitrary monoids.

Definition

. Let

M

be a monoid. A

semiautomaton

over

M

is a directed graph

G

whose edges are labeled by elements of

M

. An

automaton

over

M

is a semiautomaton over

M

, where the

vertex set

has two designated subsets

I

and

F

(not necessarily

disjoint

), where elements of

I

are called the start nodes, and elements of

F

the final nodes.

Note that if

M

=

Σ

*

for some alphabet

Σ

, then a semiautomaton

G

over

M

according to the definition given above is not necessarily a semiautomaton over

Σ

under the standard definition of a semiautomaton, since labels of the edges are words over

Σ

, not elements of

Σ

. However,

G

can be “transformed” into a “standard” semiautomaton (over

Σ

).

Definition

. Let

A

be a finite automaton over a monoid

M

. An element in

M

is said to be

accepted by

A

if it is the label of a path that begins at an initial node and end at a final node. The set of all elements of

M

accepted by

A

is denoted by

L

⁢

(

A

)

.

The following is a

generalization

of Kleene’s

theorem

.

Theorem 1

.

. A subset

R

of a monoid

M

is a

rational set

iff

R

=

L

⁢

(

A

)

for some finite automaton

A

over

M

.

References

1

S. Eilenberg,

Automata,

Languages

, and

Machines

, Vol. A

, Academic Press (1974).

Title

automaton over a monoid

Canonical

name

AutomatonOverAMonoid

Date of creation

2013-03-22 19:01:31

Last modified on

2013-03-22 19:01:31

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

7

Author

CWoo (3771)

Entry type

Definition

Classification

msc 20M35

Classification

msc 68Q70

Generated on Fri Feb 9 19:34:51 2018 by

LaTeXML
characterization of a Kleene algebra

characterization of a Kleene algebra

Let

A

be an idempotent semiring with a unary

operator

*

on

A

. The following are

equivalent

1.

a

⁢

c

+

b

≤

c

implies

a

*

⁢

b

≤

c

,

2.

a

⁢

b

≤

b

implies

a

*

⁢

b

≤

b

.

Proof.

(

1

⇒

2

)

. Assume

a

⁢

b

≤

b

. So

a

⁢

b

+

b

=

b

. Then

(

a

⁢

b

+

b

)

+

b

=

a

⁢

b

+

(

b

+

b

)

=

a

⁢

b

+

b

=

b

, which implies

a

⁢

b

+

b

≤

b

. By

1

, this means

a

*

⁢

b

≤

b

as desired.

(

2

⇒

1

)

. Assume

a

⁢

c

+

b

≤

c

. Since

0

≤

b

, we get

a

⁢

c

=

a

⁢

c

+

0

≤

a

⁢

c

+

b

≤

c

. Consequently

a

*

⁢

c

≤

c

.
∎

From the above observation, we see that we get an equivalent

definition

of a

Kleene algebra

if the axioms

a

⁢

c

+

b

≤

c

⁢

implies

⁢

a

*

⁢

b

≤

c

and

c

⁢

a

+

b

≤

c

⁢

implies

⁢

b

⁢

a

*

≤

c

are replaced by

a

⁢

b

≤

b

⁢

implies

⁢

a

*

⁢

b

≤

b

and

b

⁢

a

≤

b

⁢

implies

⁢

b

⁢

a

*

≤

b

.

Let

A

be a Kleene algebra. Some of the interesting

properties

of

*

on

A

are the following:

1.

0

*

=

1

.

2.

a

n

≤

a

*

for all non-negative integers

n

.

3.

1

*

⁢

a

≤

a

.

4.

1

*

=

1

.

5.

1

+

a

*

=

a

*

.

6.

a

≤

b

implies

a

*

≤

b

*

.

7.

a

*

⁢

a

*

=

a

*

.

8.

a

*

*

=

a

*

.

9.

1

+

a

⁢

a

*

=

a

*

.

10.

1

+

a

*

⁢

a

=

a

*

.

11.

a

⁢

c

≤

c

⁢

b

implies

a

*

⁢

c

≤

c

⁢

b

*

.

12.

c

⁢

b

≤

a

⁢

c

implies

c

⁢

b

*

≤

a

*

⁢

c

.

13.

(

a

⁢

b

)

*

⁢

a

=

a

⁢

(

b

⁢

a

)

*

.

14.

1

+

a

⁢

(

b

⁢

a

)

*

⁢

b

=

(

a

⁢

b

)

*

.

15.

If

c

=

c

2

and

1

≤

c

, then

c

*

=

c

.

16.

(

a

+

b

)

*

=

a

*

⁢

(

b

⁢

a

*

)

*

.

Remarks

.

•

Properties 9 and 10 imply that the axioms

1

+

a

⁢

a

*

≤

a

*

and

1

+

a

*

⁢

a

≤

a

*

of a Kleene algebra can be replaced by these stronger ones.

•

Though in the example of Kleene algebras formed by

regular expressions

,

a

*

=

⋃

{

a

i

∣

i

=

0

,

1

,

2

,

…

}

,

and we see that

a

*

is the least

upper bound

of the

a

i

’s, this is not a property of a general Kleene algebra. A

counterexample

of this can be found in Kozen’s article, see below. He calls a Kleene algebra

K

*

-

continuous

if

a

*

≤

b

whenever

a

i

≤

b

for any

i

∈

ℕ

∪

{

0

}

, and

a

,

b

∈

K

.

References

1

D. Kozen,

http://www.cs.cornell.edu/ kozen/papers/kacs.ps

On Kleene Algebras and Closed Semirings

(1990).

Title

characterization

of a Kleene algebra

Canonical

name

CharacterizationOfAKleeneAlgebra

Date of creation

2013-03-22 17:02:40

Last modified on

2013-03-22 17:02:40

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

12

Author

CWoo (3771)

Entry type

Definition

Classification

msc 20M35

Classification

msc 68Q70

Defines

*

-continuous

Generated on Fri Feb 9 19:35:01 2018 by

LaTeXML
generalized regular expression

generalized regular expression

Recall that a

regular expression

over an

alphabet

Σ

is a finite strings of symbols that are

elements of

Σ

, together with symbols

∅

,

∪

,

⋅

,

*

, as well as

(

and

)

, which is put together based on a set of

expression

formation rules. In a

generalized regular expression

, additional symbols

∩

and

¬

are tossed in also. Formally,

Definition

. Given an alphabet

Σ

, let

S

be the set

{

∅

,

∪

,

⋅

,

*

,

∩

,

¬

,

(

,

)

}

, considered to be

disjoint

from

Σ

. Let

X

be the smallest subset of

(

Σ

∪

S

)

*

containing the following:

•

any regular expression is in

X

,

•

if

u

,

v

∈

Y

, then

(

u

∩

v

)

,

(

¬

⁢

u

)

∈

X

.

An element of

X

is called a

generalized regular expression

over

Σ

.

Like regular expressions, every generalized regular expressions are designed to represent

languages

(it is clear that

∩

and

¬

are intended to mean set-theoretic

intersection

and

complementation

). If

u

is a generalized regular expression:

•

if

u

is regular expression, then the language represented by

u

as a generalized regular expression is

L

⁢

(

u

)

, the language represented by

u

as a regular expression;

•

if

A

is represented by

u

and

B

is represented by

v

, then

A

∩

B

is represented by

(

u

∩

v

)

•

if

A

is represented by

u

, then

Σ

*

-

A

is represented by

(

¬

⁢

u

)

.

By

induction

, it is easy to see that, given a generalized regular expression

u

, there is exactly one language represented by

u

. We denote

L

⁢

(

u

)

the language represented by

u

, and

𝒢

⁢

ℛ

the family of languages represented by generalized regular expressions.

Since

regular languages

are

closed under

intersection and complementation, generalized regular expressions in this regard are no powerful than regular expressions. The symbols

¬

and

∩

are therefore extraneous. In other words,

ℛ

=

𝒢

⁢

ℛ

,

where

ℛ

is the family of regular languages.

References

1

A. Salomaa,

Formal Languages

, Academic Press, New York (1973).

Title

generalized regular expression

Canonical

name

GeneralizedRegularExpression

Date of creation

2013-03-22 19:01:34

Last modified on

2013-03-22 19:01:34

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

5

Author

CWoo (3771)

Entry type

Definition

Classification

msc 20M35

Classification

msc 68Q70

Generated on Fri Feb 9 19:35:31 2018 by

LaTeXML
Kleene algebra

Kleene algebra

A

Kleene algebra

(

A

,

⋅

,

+

,

*

,

0

,

1

)

is an

idempotent semiring

(

A

,

⋅

,

+

,

0

,

1

)

with an additional (right-associative) unary

operator

*

, called the

Kleene star

, which

satisfies

1

+

a

⁢

a

*

≤

a

*

,

a

⁢

c

+

b

≤

c

⇒

a

*

⁢

b

≤

c

,

1

+

a

*

⁢

a

≤

a

*

,

c

⁢

a

+

b

≤

c

⇒

b

⁢

a

*

≤

c

,

for all

a

,

b

,

c

∈

A

.

For a given

alphabet

Σ

, the set of all

languages

over

Σ

, as well as the set of all

regular languages

over

Σ

, are examples of Kleene algebras. Similarly, sets of

regular expressions

(regular sets) over

Σ

are a form (or close variant) of a Kleene algebra: let

A

be the set of all regular sets over a set

Σ

of alphabets. Then

A

is a Kleene algebra if we identify

∅

as

0

, the singleton containing the

empty string

λ

as

1

,

concatenation

operation

as

⋅

, the union operation as

+

, and the Kleene star operation as

*

. For example, let

a

be a set of regular expression, then

a

*

=

{

λ

}

∪

a

∪

a

2

∪

⋯

∪

a

n

∪

⋯

,

so that

a

⁢

a

*

=

a

∪

a

2

∪

⋯

∪

a

n

∪

⋯

.

Adding

1

on both sides and we have

1

+

a

⁢

a

*

=

{

λ

}

∪

a

⁢

a

*

=

{

λ

}

∪

a

∪

a

2

∪

⋯

∪

a

n

∪

⋯

=

a

*

.

The other conditions are checked similarly.

Remark

. There is another notion of a Kleene algebra, which arises from lattices. For more detail, see here (

http://planetmath.org/KleeneAlgebra2

).

Title

Kleene algebra

Canonical name

KleeneAlgebra

Date of creation

2013-03-22 12:27:51

Last modified on

2013-03-22 12:27:51

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

10

Author

CWoo (3771)

Entry type

Definition

Classification

msc 20M35

Classification

msc 68Q70

Related topic

KleeneStar

Related topic

Semiring

Related topic

RegularExpression

Related topic

RegularLanguage

Related topic

KleeneAlgebra2

Generated on Fri Feb 9 19:35:33 2018 by

LaTeXML
Kleene star

Kleene star

If

Σ

is an

alphabet

(a set of symbols), then the

Kleene star

of

Σ

, denoted

Σ

*

, is the set of all strings of finite length consisting of symbols in

Σ

, including the

empty string

λ

.

*

is also called the

asterate

.

If

S

is a set of strings, then the Kleene star of

S

,
denoted

S

*

, is the smallest

superset

of

S

that contains

λ

and
is

closed under

the string concatenation

operation

. That is,

S

*

is the
set of all strings that can be generated by concatenating zero or more strings in

S

.

The

definition

of Kleene star can be generalized so that it operates on any
monoid

(

M

,

+

+

)

, where

+

⁣

+

is a

binary operation

on the set

M

.
If

e

is the

identity element

of

(

M

,

+

+

)

and

S

is a subset of

M

, then

S

*

is the smallest superset of

S

that
contains

e

and is closed under

+

⁣

+

.

Examples

•

∅

*

=

{

λ

}

, since there are no strings of finite length consisting of symbols in

∅

, so

λ

is the only

element

in

∅

*

.

•

If

E

=

{

λ

}

, then

E

*

=

E

, since

λ

⁢

a

=

a

⁢

λ

=

a

by definition, so

λ

⁢

λ

=

λ

.

•

If

A

=

{

a

}

, then

A

*

=

{

λ

,

a

,

a

⁢

a

,

a

⁢

a

⁢

a

,

…

}

.

•

If

Σ

=

{

a

,

b

}

, then

Σ

*

=

{

λ

,

a

,

b

,

a

⁢

a

,

a

⁢

b

,

b

⁢

a

,

b

⁢

b

,

a

⁢

a

⁢

a

,

…

}

•

If

S

=

{

a

⁢

b

,

c

⁢

d

}

, then

S

*

=

{

λ

,

a

⁢

b

,

c

⁢

d

,

a

⁢

b

⁢

a

⁢

b

,

a

⁢

b

⁢

c

⁢

d

,

c

⁢

d

⁢

a

⁢

b

,

c

⁢

d

⁢

c

⁢

d

,

a

⁢

b

⁢

a

⁢

b

⁢

a

⁢

b

,

…

}

For any set

S

,

S

*

is the

free monoid

generated by

S

.

Remark

. There is an associated operation, called the

Kleene plus

, is defined for any set

S

, such that

S

+

is the smallest set containing

S

such that

S

+

is closed under the

concatenation

. In other words,

S

+

=

S

*

-

{

λ

}

.

Title

Kleene star

Canonical

name

KleeneStar

Date of creation

2013-03-22 12:26:58

Last modified on

2013-03-22 12:26:58

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

9

Author

CWoo (3771)

Entry type

Definition

Classification

msc 20M35

Classification

msc 68Q70

Synonym

asterate

Related topic

Alphabet

Related topic

String

Related topic

RegularExpression

Related topic

KleeneAlgebra

Related topic

Language

Related topic

Convolution2

Related topic

WeightStrings

Related topic

WeightEnumerator

Defines

Kleene plus

Generated on Fri Feb 9 19:35:35 2018 by

LaTeXML
star height

star height

The

star height

of a

regular expression

p

over an

alphabet

Σ

, denoted by

ht

⁡

(

p

)

, is defined recursively as follows:

1.

ht

⁡

(

∅

)

=

ht

⁡

(

a

)

=

0

for any

a

∈

Σ

;

2.

ht

⁡

(

p

∪

q

)

=

ht

⁡

(

p

⁢

q

)

=

max

⁡

(

ht

⁡

(

p

)

,

ht

⁡

(

q

)

)

;

3.

ht

⁡

(

p

*

)

=

ht

⁡

(

p

)

+

1

.

Let

Σ

=

{

a

,

b

,

c

}

. The following

expressions

have star height

0

,

1

,

2

,

3

:

(

a

∪

b

)

⁢

c

a

*

⁢

b

*

(

a

*

⁢

b

)

*

⁢

c

(

(

a

⁢

b

*

⁢

a

∪

c

)

*

⁢

b

)

*

Since any regular expression

p

describes a

regular language

L

⁢

(

p

)

, we may extend the

definition

of star height to regular languages. However, since a regular language may be described by more than one regular expressions, we need to be a little careful:

The

star height

of a regular language

L

is the least integer

i

such that

L

may be described by a regular expression of star height

i

. In other words:

ht

⁡

(

L

)

=

min

⁡

{

h

⁢

(

p

)

∣

L

=

L

⁢

(

p

)

⁢

,

⁢

p

∈

R

⁢

(

Σ

)

}

,

where

R

⁢

(

Σ

)

is the set of all regular expressions over

Σ

.

Remarks

.

•

Any regular language over a

singleton

alphabet has star height at most 1, which can be proved by the pumping lemma for regular languages.

•

If the alphabet

Σ

consists of at least two letters, then for every positive integer

n

, there is a regular language whose star height is

n

. In fact, it can be shown that if

|

Σ

|

=

2

, then for every

n

, there is a regular language

L

over

Σ

such that

ht

⁡

(

L

)

=

n

.

•

It was an

open question

whether an

algorithm

exists for determining

ht

⁡

(

L

)

for an arbitrary regular language

L

. In 2005, the question was resolved by D. Kirsten in the

positive

, and the algorithm is that of a

non-deterministic finite automaton

.

•

We may also define star height on

generalized regular expressions

. For a regular language

L

, define

ht

g

⁡

(

L

)

=

min

⁡

{

h

⁢

(

p

)

∣

L

=

L

⁢

(

p

)

⁢

,

⁢

p

∈

R

g

⁢

(

Σ

)

}

, where

R

g

⁢

(

Σ

)

is the set of all generalized regular expressions. It is clear that

ht

g

⁡

(

L

)

≤

ht

⁡

(

L

)

. However, it is still an open question whether, for every integer

n

, there is a regular language

L

with

ht

g

⁡

(

L

)

=

n

.

A star-free language has star height

0

with respect to

representations

by generalized regular expressions, but may be positive with respect to representations by regular expressions, for example

L

=

{

a

⁢

b

}

*

.

References

1

A. Salomaa,

Formal Languages

, Academic Press, New York (1973).

2

A. Salomaa,

Jewels of Formal Language Theory

, Computer Science Press (1981).

Title

star height

Canonical

name

StarHeight

Date of creation

2013-03-22 18:57:55

Last modified on

2013-03-22 18:57:55

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

9

Author

CWoo (3771)

Entry type

Definition

Classification

msc 20M35

Classification

msc 68Q70

Related topic

StarFree

Generated on Fri Feb 9 19:35:59 2018 by

LaTeXML
a semilattice is a commutative band

a semilattice is a commutative band

This note explains how a

semilattice

is the same as a

commutative

band.

Let

S

be a semilattice, with

partial order

<

and each pair of

elements

x

and

y

having a

greatest lower bound

x

∧

y

.
Then it is easy to see that the

operation

∧

defines a

binary operation

on

S

which makes it a commutative semigroup, and that every element is

idempotent

since

x

∧

x

=

x

.

Conversely

, if

S

is such a

semigroup

, define

x

≤

y

iff

x

=

x

⁢

y

. Again, it is easy to see that this defines a partial order on

S

, and that greatest lower bounds exist with respect to this partial order, and that in fact

x

∧

y

=

x

⁢

y

.

Title

a semilattice is a commutative band

Canonical name

ASemilatticeIsACommutativeBand

Date of creation

2013-03-22 12:57:28

Last modified on

2013-03-22 12:57:28

Owner

mclase (549)

Last modified by

mclase (549)

Numerical id

6

Author

mclase (549)

Entry type

Proof

Classification

msc 20M99

Classification

msc 06A12

Related topic

Lattice

Generated on Fri Feb 9 19:36:10 2018 by

LaTeXML
adjoining an identity to a semigroup

adjoining an identity to a semigroup

It is possible to formally adjoin an

identity element

to any

semigroup

to make it into a monoid.

Suppose

S

is a semigroup without an

identity

, and consider the set

S

∪

{

1

}

where

1

is a symbol not in

S

. Extend the semigroup

operation

from

S

to

S

∪

{

1

}

by additionally defining:

s

⋅

1

=

s

=

1

⋅

s

,

for all

⁢

s

∈

S

∪

1

It is easy to verify that this defines a semigroup (

associativity

is the only thing that needs to be checked).

As a matter of notation, it is customary to write

S

1

for the semigroup

S

with an identity adjoined in this manner, if

S

does not already have one, and to agree that

S

1

=

S

, if

S

does already have an identity.

Despite the simplicity of this construction, however, it rarely allows one to simplify a problem by considering monoids instead of semigroups. As soon as one starts to look at the

structure

of the semigroup, it is almost invariably the case that one needs to consider

subsemigroups

and ideals of the semigroup which do not contain the identity.

Title

adjoining an identity to a semigroup

Canonical name

AdjoiningAnIdentityToASemigroup

Date of creation

2013-03-22 13:01:19

Last modified on

2013-03-22 13:01:19

Owner

mclase (549)

Last modified by

mclase (549)

Numerical id

5

Author

mclase (549)

Entry type

Definition

Classification

msc 20M99

Related topic

Semigroup

Related topic

Monoid

Generated on Fri Feb 9 19:36:11 2018 by

LaTeXML
band

band

A

band

is a

semigroup

in which every

element

is

idempotent

.

In semigroup theory, a

commutative

band is called a

semilattice

. This is

http://planetmath.org/node/3320

equivalent

to the defintion of a semilattice from the theory of

partially ordered sets

.

Title

band

Canonical name

Band

Date of creation

2013-03-22 12:57:26

Last modified on

2013-03-22 12:57:26

Owner

mclase (549)

Last modified by

mclase (549)

Numerical id

5

Author

mclase (549)

Entry type

Definition

Classification

msc 20M99

Related topic

Semigroup

Generated on Fri Feb 9 19:36:13 2018 by

LaTeXML