aarjaneiro/planetmath huggingface dataset:
Wagner congruence

Wagner congruence

Let

ρ

~

X

⊆

(

X

∪

X

-

1

)

+

be the

binary relation

on the

free semigroup with involution

(

X

∪

X

-

1

)

+

defined by

ρ

~

X

=

{

(

w

⁢

w

-

1

⁢

w

,

w

)

,

(

w

⁢

w

-

1

⁢

v

⁢

v

-

1

,

v

⁢

v

-

1

⁢

w

⁢

w

-

1

)

|

v

,

w

∈

(

X

∪

X

-

1

)

+

}

.

The

Wagner congruence

on

X

is the

congruence

ρ

X

generated by

ρ

~

X

, i.e.

ρ

X

=

(

ρ

~

X

)

c

.

A well known result of

inverse semigroups

theory says that the

quotient

FIS

⁢

(

X

)

=

(

X

∪

X

-

1

)

+

/

ρ

X

is an inverse semigroup. Moreover

FIS

⁢

(

X

)

is the

Free Inverse Semigroup

on

X

, in the sense that it resolve the following

universal

mapping

problem: given an inverse semigroup

S

and a map

Φ

:

X

→

S

, a unique inverse semigroups

homomorphism

Φ

¯

:

FIS

⁢

(

X

)

→

S

exists such that the following

diagram

commutes:

\xymatrix

⁢

&

⁢

X

⁢

\ar

⁢

[

r

]

ι

⁢

\ar

⁢

[

d

]

Φ

⁢

&

⁢

FIS

⁢

(

X

)

⁢

\ar

⁢

[

d

⁢

l

]

Φ

¯

⁢

&

⁢

S

⁢

&

where

ι

:

X

→

FIS

⁢

(

X

)

is the

projection

to the quotient, i.e.

ι

⁢

(

x

)

=

[

x

]

ρ

X

. It is well known from

universal algebra

that

FIS

⁢

(

X

)

is unique up to

isomorphisms

.

In analogous way, using the free monoid with involution

(

X

∪

X

-

1

)

∗

instead of the free semigroup with involution

(

X

∪

X

-

1

)

+

, we obtain the

inverse

monoid

FIM

⁢

(

X

)

=

(

X

∪

X

-

1

)

∗

/

ρ

X

,

that is the

Free Inverse Monoid

on

X

.

References

1

N. Petrich,

Inverse Semigroups

, Wiley, New York, 1984.

2

V.V. Wagner,

Generalized Groups

, Dokl. Akad. Nauk SSSR 84 (1952), 1119-1122.

Title

Wagner congruence

Canonical name

WagnerCongruence

Date of creation

2013-03-22 16:11:07

Last modified on

2013-03-22 16:11:07

Owner

Mazzu (14365)

Last modified by

Mazzu (14365)

Numerical id

15

Author

Mazzu (14365)

Entry type

Definition

Classification

msc 20M05

Classification

msc 20M18

Defines

Wagner congruence

Defines

free inverse semigroup

Defines

free inverse monoid

Generated on Fri Feb 9 19:33:38 2018 by

LaTeXML
cancellative semigroup

cancellative semigroup

Let

S

be a

semigroup

.

S

is

left cancellative

if, for all

a

,

b

,

c

∈

S

,

a

⁢

b

=

a

⁢

c

⇒

b

=

c

S

is

right cancellative

if, for all

a

,

b

,

c

∈

S

,

b

⁢

a

=

c

⁢

a

⇒

b

=

c

S

is

cancellative

if it is both left and right cancellative.

1

Relationship to some other types of semigroup

This is a generalisation of groups, and in fact being cancellative is a necessary condition for a semigroup to be embeddable in a group.

Note that a non-empty semigroup is a group if and only if it is cancellative and

regular

.

S

is

weakly cancellative

if, for all

a

,

b

,

c

∈

S

,

(

a

b

=

a

c

&

b

a

=

c

a

)

⇒

b

=

c

A semigroup is

completely simple

if and only if it is weakly cancellative and regular.

2

Individual elements

An element

x

∈

S

is called

left cancellative

if, for all

b

,

c

∈

S

,

x

⁢

b

=

x

⁢

c

⇒

b

=

c

An element

x

∈

S

is called

right cancellative

if, for all

b

,

c

∈

S

,

b

⁢

x

=

c

⁢

x

⇒

b

=

c

Title

cancellative semigroup

Canonical

name

CancellativeSemigroup

Date of creation

2013-03-22 14:25:09

Last modified on

2013-03-22 14:25:09

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

9

Author

yark (2760)

Entry type

Definition

Classification

msc 20M10

Synonym

cancellation semigroup

Related topic

CancellationIdeal

Defines

cancellative

Defines

weakly cancellative

Defines

left cancellative

Defines

right cancellative

Defines

weakly cancellative semigroup

Defines

left cancellative semigroup

Defines

right cancellative semigroup

Generated on Fri Feb 9 19:33:40 2018 by

LaTeXML
completely simple semigroup

completely simple semigroup

Let

S

be a

semigroup

. An

idempotent

e

∈

S

is

primitive

if for every other idempotent

f

∈

S

,

e

⁢

f

=

f

⁢

e

=

f

≠

0

⇒

e

=

f

A semigroup

S

(without zero) is

completely

if it is simple and contains a primitive idempotent.

A semigroup

S

is

completely

0

-simple

if it is

0

-simple (

http://planetmath.org/SimpleSemigroup

) and contains a primitive idempotent.

Completely simple and completely

0

-

simple semigroups

maybe characterised by the Rees Theorem (

[

Ho95

]

, Theorem 3.2.3).

Note:

A semigroup (without zero) is completely simple if and only if it is

regular

and

weakly cancellative

.

A simple semigroup (without zero) is completely simple if and only if it is

completely regular

.

A

0

-simple semigroup is completely

0

-simple if and only if it is group-bound.

References

Ho95

Howie, John M.

Fundamentals of Semigroup Theory

. Oxford University Press, 1995.

Title

completely simple semigroup

Canonical name

CompletelySimpleSemigroup

Date of creation

2013-03-22 14:35:24

Last modified on

2013-03-22 14:35:24

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

8

Author

mathcam (2727)

Entry type

Definition

Classification

msc 20M10

Defines

primitive

Defines

completely

0

-simple

Defines

completely simple

Generated on Fri Feb 9 19:33:42 2018 by

LaTeXML
defect theorem

defect theorem

Let

A

be an arbitrary set,
let

A

∗

be the

free monoid

on

A

,
and let

X

be a finite subset of

A

∗

which is not a code.
Then the

free hull

H

of

X

is itself finite
and

|

H

|

<

|

X

|

.

Observe how from the

defect theorem

follows that the

equation

x

m

=

y

n

over

A

∗

has only the

trivial

solutions where

x

and

y

are powers of the same word:
in fact,

x

m

=

y

n

iff

{

x

,

y

}

is not a code.

Proof.

It is

sufficient

to prove the thesis in the case
when

X

does not contain the

empty word

on

A

.

Define

f

:

X

→

H

such that

f

⁢

(

x

)

is the only

h

∈

H

such that

x

∈

h

⁢

H

∗

:

f

is well defined because of the choice of

X

and

H

.
Since

X

is finite, the thesis shall be established
once we prove that

f

is

surjective

but not

injective

.

By

hypothesis

,

X

is not a code.
Then there exists an equation

x

1

⁢

⋯

⁢

x

n

=

x

1

′

⁢

⋯

⁢

x

m

′

over

X

with a nontrivial solution:
it is not restrictive to suppose that

x

1

≠

x

1

′

.
Since however the factorization

over

H

is unique,
the

h

such that

x

1

∈

h

⁢

H

∗

is the same as the

h

′

such that

x

1

′

∈

h

′

⁢

H

∗

:
then

f

⁢

(

x

1

)

=

f

⁢

(

x

1

′

)

with

x

1

≠

x

1

′

,
so

f

is not injective.

Now suppose, for the sake of

contradiction

,
that

h

∈

H

∖

f

⁢

(

X

)

exists.
Let

K

=

(

H

∖

{

h

}

)

⁢

h

∗

.
Then any equation

k

1

⁢

⋯

⁢

k

n

=

k

1

′

⁢

⋯

⁢

k

m

′

,

k

1

,

⋯

,

k

n

,

k

1

′

,

…

,

k

m

′

∈

K

(1)

can be rewritten as

h

1

⁢

h

r

1

⁢

⋯

⁢

h

n

⁢

h

r

n

=

h

1

′

⁢

h

s

1

⁢

⋯

⁢

h

m

′

⁢

h

s

m

,

h

1

,

…

,

h

n

,

h

1

′

,

…

,

h

m

′

∈

H

∖

{

h

}

,

with

k

i

=

h

i

⁢

h

r

i

,

k

i

′

=

h

i

′

⁢

h

s

i

:
this is an equation over

H

,
and as such, has only the trivial solution

n

=

m

,

h

i

=

h

i

′

,

r

i

=

s

i

for all

i

.
This implies that (

1

) only has trivial solutions over

K

:
by the

characterization of free submonoids

,

K

is a code.
However,

X

⊆

K

∗

because no element of

x

“starts with

h

”,
and

K

∗

is a

proper subset

of

H

∗

because the former does not contain

h

and the latter does.
Then

K

∗

is a

free submonoid

of

A

∗

which contains

X

and is properly

contained

in

H

∗

,
against the

definition

of

H

as the free hull of

X

.

References

1

M. Lothaire.

Combinatorics on words.

Cambridge University Press 1997.

Title

defect theorem

Canonical

name

DefectTheorem

Date of creation

2013-03-22 18:21:43

Last modified on

2013-03-22 18:21:43

Owner

Ziosilvio (18733)

Last modified by

Ziosilvio (18733)

Numerical id

6

Author

Ziosilvio (18733)

Entry type

Theorem

Classification

msc 20M10

Classification

msc 20M05

Defines

x

m

=

y

n

has only trivial solutions over

A

∗

Generated on Fri Feb 9 19:33:45 2018 by

LaTeXML
existence of maximal semilattice decomposition

existence of maximal semilattice decomposition

Let

S

be a

semigroup

. A

maximal semilattice decomposition

for

S

is a

surjective

homomorphism

ϕ

:

S

→

Γ

onto a

semilattice

Γ

with the property that any other semilattice

decomposition

factors through

ϕ

. So if

ϕ

′

:

S

→

Γ

′

is any other semilattice decomposition of

S

, then there is a homomorphism

Γ

→

Γ

′

such that the following

diagram

commutes:

\xymatrix

S

\ar

[

r

]

ϕ

\ar

[

d

r

]

ϕ

′

&

Γ

\ar

@

-

-

>

[

d

]

&

Γ

′

Proposition

.

Every semigroup has a maximal semilattice decomposition.

Proof.

Recall that each semilattice decompostion determines a

semilattice congruence

. If

{

ρ

i

∣

i

∈

I

}

is the family of all semilattice congruences on

S

, then define

ρ

=

⋂

i

∈

I

ρ

i

. (Here, we consider the

congruences

as subsets of

S

×

S

, and take their

intersection

as sets.)

It is easy to see that

ρ

is also a semilattice congruence, which is

contained

in all other semilattice congruences.

Therefore each of the homomorphisms

S

→

S

/

ρ

i

factors through

S

→

S

/

ρ

.
∎

Title

existence of maximal semilattice decomposition

Canonical name

ExistenceOfMaximalSemilatticeDecomposition

Date of creation

2013-03-22 13:07:12

Last modified on

2013-03-22 13:07:12

Owner

mclase (549)

Last modified by

mclase (549)

Numerical id

5

Author

mclase (549)

Entry type

Result

Classification

msc 20M10

Defines

minimal semilattice congruence

Defines

maximal semilattice decomposition

Generated on Fri Feb 9 19:33:47 2018 by

LaTeXML
free semigroup

free semigroup

Let

X

be a set.
We define the power of

X

in a language-theoretical manner as

X

n

=

{

x

1

⁢

x

2

⁢

…

⁢

x

n

∣

x

j

∈

X

⁢

for all

⁢

j

∈

{

1

,

…

,

n

}

}

for all

n

∈

ℕ

∖

{

0

}

, and

X

0

=

{

ε

}

,

where

ε

∉

X

.
Note that the set

X

is not necessarily an

alphabet

,
that is, it may be

infinite

; for example, we may choose

X

=

ℝ

.

We define the sets

X

+

and

X

*

as

X

+

=

⋃

n

∈

ℕ

∖

{

0

}

X

n

and

X

*

=

⋃

n

∈

ℕ

X

n

=

X

+

∪

{

ε

}

.

The elements of

X

*

are called

words

on

X

,
and

ε

is called the

empty word

on

X

.

We define the

juxtaposition

of two words

v

,

w

∈

X

*

as

v

⁢

w

=

v

1

⁢

v

2

⁢

…

⁢

v

n

⁢

w

1

⁢

w

2

⁢

…

⁢

w

m

,

where

v

=

v

1

⁢

v

2

⁢

…

⁢

v

n

and

w

=

w

1

⁢

w

2

⁢

…

⁢

w

m

,
with

v

i

,

w

j

∈

X

for each

i

and

j

.
It is easy to see that the juxtaposition is

associative

,
so if we equip

X

+

and

X

*

with it
we obtain respectively a

semigroup

and a monoid.
Moreover,

X

+

is the

free semigroup

on

X

and

X

*

is the

free monoid

on

X

,
in the sense that they solve the following

universal

mapping

problem:
given a semigroup

S

(resp. a monoid

M

)
and a map

Φ

:

X

→

S

(resp.

Φ

:

X

→

M

),
a semigroup homomorphism

Φ

¯

:

X

+

→

S

(resp. a

monoid homomorphism

Φ

¯

:

X

*

→

M

)
exists such that the following

diagram

commutes:

\xymatrix

⁢

&

⁢

X

⁢

\ar

⁢

[

r

]

ι

⁢

\ar

⁢

[

d

]

Φ

⁢

&

⁢

X

+

⁢

\ar

⁢

[

d

⁢

l

]

Φ

¯

⁢

&

⁢

S

⁢

&

(resp.

\xymatrix

⁢

&

⁢

X

⁢

\ar

⁢

[

r

]

ι

⁢

\ar

⁢

[

d

]

Φ

⁢

&

⁢

X

*

⁢

\ar

⁢

[

d

⁢

l

]

Φ

¯

⁢

&

⁢

M

⁢

&

), where

ι

:

X

→

X

+

(resp.

ι

:

X

→

X

*

)
is the

inclusion map

.
It is well known from

universal algebra

that

X

+

and

X

*

are unique up to

isomorphism

.

References

1

J.M. Howie,

Fundamentals of Semigroup Theory,
Oxford University Press, Oxford, 1991.

Title

free semigroup

Canonical name

FreeSemigroup

Date of creation

2013-03-22 16:11:41

Last modified on

2013-03-22 16:11:41

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

15

Author

yark (2760)

Entry type

Definition

Classification

msc 20M10

Classification

msc 20M05

Related topic

Word

Defines

word

Defines

empty word

Defines

free semigroup

Defines

free monoid

Generated on Fri Feb 9 19:33:49 2018 by

LaTeXML
free semigroup with involution

free semigroup with involution

Let

X

,

X

‡

be two

disjoint sets

in

bijective

correspondence given by the map

:

‡

X

→

X

‡

. Denote by

Y

=

X

∐

X

‡

(here we use

∐

instead of

∪

to remind that the union is actually a

disjoint union

) and by

Y

+

the

free semigroup

on

Y

. We can extend the map

‡

to an

involution

:

‡

Y

+

→

Y

+

on

Y

+

in the following way: given

w

∈

Y

+

, we have

w

=

w

1

⁢

w

2

⁢

…

⁢

w

k

for some letters

w

i

∈

Y

; then we define

w

‡

=

w

k

‡

⁢

w

k

-

1

‡

⁢

…

⁢

w

2

‡

⁢

w

1

‡

.

It is easily verified that this is the unique way to extend

‡

to an involution on

Y

. Thus, the

semigroup

(

X

∐

X

‡

)

+

with the involution

‡

is a semigroup with involution. Moreover, it is the

free semigroup with involution

on

X

, in the sense that it solves the following universal problem: given a semigroup with involution

S

and a map

Φ

:

X

→

S

, a semigroup homomorphism

Φ

¯

:

(

X

∐

X

‡

)

+

→

S

exists such that the following

diagram

commutes:

\xymatrix

⁢

&

⁢

X

⁢

\ar

⁢

[

r

]

ι

⁢

\ar

⁢

[

d

]

Φ

⁢

&

⁢

(

X

∐

X

‡

)

+

⁢

\ar

⁢

[

d

⁢

l

]

Φ

¯

⁢

&

⁢

S

⁢

&

where

ι

:

X

→

(

X

∐

X

‡

)

+

is the

inclusion map

. It is well known from

universal algebra

that

(

X

∐

X

‡

)

+

is unique up to

isomorphisms

.

If we use

Y

*

instead of

Y

+

, where

Y

*

=

Y

+

∪

{

ε

}

and

ε

is the

empty word

(i.e. the

identity

of the monoid

Y

*

), we obtain a monoid with involution

(

X

∐

X

‡

)

*

that is the

free monoid with involution

on

X

.

Title

free semigroup with involution

Canonical name

FreeSemigroupWithInvolution

Date of creation

2013-03-22 16:11:30

Last modified on

2013-03-22 16:11:30

Owner

Mazzu (14365)

Last modified by

Mazzu (14365)

Numerical id

8

Author

Mazzu (14365)

Entry type

Example

Classification

msc 20M10

Defines

free semigroup with involution

Defines

free monoid with involution

Generated on Fri Feb 9 19:33:51 2018 by

LaTeXML
free submonoid

free submonoid

Let

A

be an arbitrary set,
let

A

∗

be the

free monoid

on

A

,
and let

e

be the

identity element

(

empty word

) of

A

∗

.

Let

M

be a

submonoid

of

A

∗

.
The

minimal

generating set

of

M

is

mgs

⁢

(

M

)

=

(

M

∖

{

e

}

)

∖

(

M

∖

{

e

}

)

2

.

(1)

Shortly,

mgs

⁢

(

M

)

is the set of all the

nontrivial elements

of

M

that cannot be “reconstructed” as

products

of elements of

M

.
It is straightforward that

1.

(

mgs

⁢

(

M

)

)

∗

=

M

, and

2.

if

S

⊆

A

∗

and

M

⊆

S

∗

,
then

mgs

⁢

(

M

)

⊆

S

.

We say that

M

is a

free submonoid

of

A

∗

if it is

isomorphic

(as a monoid)
to a free monoid

B

∗

for some set

B

.
A set

K

⊆

A

∗

such that

K

=

mgs

⁢

(

M

)

for some free submonoid

M

of

A

∗

is also called a

code

.

Title

free submonoid

Canonical

name

FreeSubmonoid

Date of creation

2013-03-22 18:21:36

Last modified on

2013-03-22 18:21:36

Owner

Ziosilvio (18733)

Last modified by

Ziosilvio (18733)

Numerical id

5

Author

Ziosilvio (18733)

Entry type

Definition

Classification

msc 20M10

Classification

msc 20M05

Defines

minimal generating set of a submonoid

Generated on Fri Feb 9 19:33:52 2018 by

LaTeXML
I-semigroup

I-semigroup

An

I

-

semigroup

[resp.

I

-monoid

] is a semigroup

S

[resp. a monoid

M

] with a

unary operation

x

↦

x

-

1

defined on

S

[resp. on

M

] such that for each

x

,

y

∈

S

[resp. for each

x

,

y

∈

M

]

(

x

-

1

)

-

1

=

x

,

x

=

x

⁢

x

-

1

⁢

x

.

Notice that

x

-

1

⁢

x

⁢

x

-

1

=

x

-

1

⁢

(

x

-

1

)

-

1

⁢

x

-

1

=

x

-

1

,

so

x

-

1

is an

inverse

of

x

.

The class of

I

-semigroups [resp.

I

-monoids] strictly contains the class of inverse semigroups [resp. inverse monoids]. In fact, the class of inverse semigroups [resp. inverse monoids] is precisely the class of

I

-

semigroups with involution

[resp.

I

-monoids with involution], i.e. the class of

I

-semigroups [resp.

I

-monoids] in which the unary operation

-

1

is also an involution.

References

1

J.M. Howie,

Fundamentals of Semigroup Theory

, Oxford University Press, Oxford, 1991.

Title

I-semigroup

Canonical name

Isemigroup

Date of creation

2013-03-22 16:11:27

Last modified on

2013-03-22 16:11:27

Owner

Mazzu (14365)

Last modified by

Mazzu (14365)

Numerical id

5

Author

Mazzu (14365)

Entry type

Definition

Classification

msc 20M10

Related topic

SemigroupWithInvolution

Defines

I-semigroup

Defines

I-monoid

Generated on Fri Feb 9 19:33:54 2018 by

LaTeXML
semigroup with involution

semigroup with involution

An

involution

on a semigroup

S

[on a monoid

M

] is a

unary operation

x

↦

x

*

defined on

S

[resp. on

M

] such that for each

x

,

y

∈

S

[resp. for each

x

,

y

∈

M

]

(

x

*

)

*

=

x

,

(

x

⁢

y

)

*

=

y

*

⁢

x

*

.

With this added internal operation, the semigroup

S

[resp. the monoid

M

] becomes a

semigroup with involution

or

*

-semigroup

[resp. a

monoid with involution

or

*

-monoid

].

References

1

J.M. Howie,

Fundamentals of Semigroup Theory

, Oxford University Press, Oxford, 1991.

Title

semigroup with involution

Canonical name

SemigroupWithInvolution

Date of creation

2013-03-22 16:11:24

Last modified on

2013-03-22 16:11:24

Owner

Mazzu (14365)

Last modified by

Mazzu (14365)

Numerical id

7

Author

Mazzu (14365)

Entry type

Definition

Classification

msc 20M10

Synonym

*-semigroup

Related topic

ISemigroup

Related topic

ThueSystem

Defines

involution

Defines

semigroup with involution

Defines

monoid with involution

Generated on Fri Feb 9 19:33:56 2018 by

LaTeXML
semilattice decomposition of a semigroup

semilattice decomposition of a semigroup

A

semigroup

S

has a

semilattice

decomposition

if we can write

S

=

⋃

γ

∈

Γ

S

γ

as a

disjoint union

of

subsemigroups

,

indexed by

elements of a semilattice

Γ

, with the additional condition that

x

∈

S

α

and

y

∈

S

β

implies

x

⁢

y

∈

S

α

⁢

β

.

Semilattice decompositions arise from homomorphims of semigroups onto semilattices. If

ϕ

:

S

→

Γ

is a

surjective

homomorphism

, then it is easy to see that we get a semilattice decomposition by putting

S

γ

=

ϕ

-

1

⁢

(

γ

)

for each

γ

∈

Γ

.

Conversely

, every semilattice decomposition defines a map from

S

to the indexing set

Γ

which is easily seen to be a homomorphism.

A third way to look at semilattice decompositions is to consider the

congruence

ρ

defined by the homomorphism

ϕ

:

S

→

Γ

. Because

Γ

is a semilattice,

ϕ

⁢

(

x

2

)

=

ϕ

⁢

(

x

)

for all

x

, and so

ρ

satisfies

the constraint that

x

⁢

ρ

⁢

x

2

for all

x

∈

S

. Also,

ϕ

⁢

(

x

⁢

y

)

=

ϕ

⁢

(

y

⁢

x

)

so that

x

⁢

y

⁢

ρ

⁢

y

⁢

x

for all

x

,

y

∈

S

.
A congruence

ρ

which satisfies these two conditions is called a

semilattice congruence

.

Conversely, a semilattice congruence

ρ

on

S

gives rise to a homomorphism from

S

to a semilattice

S

/

ρ

. The

ρ

-classes are the

components

of the decomposition.

Title

semilattice decomposition of a semigroup

Canonical name

SemilatticeDecompositionOfASemigroup

Date of creation

2013-03-22 13:07:09

Last modified on

2013-03-22 13:07:09

Owner

mclase (549)

Last modified by

mclase (549)

Numerical id

6

Author

mclase (549)

Entry type

Definition

Classification

msc 20M10

Defines

semilattice congruence

Generated on Fri Feb 9 19:33:58 2018 by

LaTeXML
simple semigroup

simple semigroup

Let

S

be a

semigroup

. If

S

has no ideals other than itself, then

S

is said to be

simple

.

If

S

has no

left ideals

[resp.

right ideals

] other than itself, then

S

is said to be

left simple

[resp.

right simple

].

Right simple and left simple are stronger conditions than simple.

A semigroup

S

is left simple if and only if

S

⁢

a

=

S

for all

a

∈

S

.
A semigroup is both left and right simple if and only if it is a group.

If

S

has a

zero element

θ

, then

0

=

{

θ

}

is always an ideal of

S

, so

S

is not simple (unless it has only one element). So in studying semigroups with a zero, a slightly weaker

definition

is required.

Let

S

be a semigroup with a zero. Then

S

is

zero simple

, or

0

-simple, if the following conditions hold:

•

S

2

≠

0

•

S

has no ideals except

0

and

S

itself

The condition

S

2

=

0

really only eliminates one semigroup: the 2-element

null semigroup

. Excluding this semigroup makes parts of the structure theory of semigroups cleaner.

Title

simple semigroup

Canonical

name

SimpleSemigroup

Date of creation

2013-03-22 13:05:59

Last modified on

2013-03-22 13:05:59

Owner

mclase (549)

Last modified by

mclase (549)

Numerical id

7

Author

mclase (549)

Entry type

Definition

Classification

msc 20M10

Defines

simple

Defines

zero simple

Defines

right simple

Defines

left simple

Generated on Fri Feb 9 19:33:59 2018 by

LaTeXML
ideal

ideal

Let

S

be a

semigroup

. An

ideal

of

S

is a non-empty subset of

S

which is

closed under

multiplication

on either side by elements of

S

. Formally,

I

is an ideal of

S

if

I

is non-empty, and for all

x

∈

I

and

s

∈

S

, we have

s

⁢

x

∈

I

and

x

⁢

s

∈

I

.

One-sided ideals

are defined similarly. A non-empty subset

A

of

S

is a

left ideal

(resp.

right ideal

) of

S

if for all

a

∈

A

and

s

∈

S

, we have

s

⁢

a

∈

A

(resp.

a

⁢

s

∈

A

).

A

principal left ideal

of

S

is a

left ideal generated by

a single

element

. If

a

∈

S

, then the principal left ideal of

S

generated by

a

is

S

1

⁢

a

=

S

⁢

a

∪

{

a

}

. (The notation

S

1

is explained here (

http://planetmath.org/AdjoiningAnIdentityToASemigroup3

).)

Similarly, the

principal right ideal

generated by

a

is

a

⁢

S

1

=

a

⁢

S

∪

{

a

}

.

The notation

L

⁢

(

a

)

and

R

⁢

(

a

)

are also common for the principal left and right

ideals generated by

a

respectively.

A

principal ideal

of

S

is an ideal generated by a single element. The ideal generated by

a

is

S

1

⁢

a

⁢

S

1

=

S

⁢

a

⁢

S

∪

S

⁢

a

∪

a

⁢

S

∪

{

a

}

.

The notation

J

⁢

(

a

)

=

S

1

⁢

a

⁢

S

1

is also common.

Title

ideal

Canonical name

Ideal1

Date of creation

2013-03-22 13:05:43

Last modified on

2013-03-22 13:05:43

Owner

mclase (549)

Last modified by

mclase (549)

Numerical id

8

Author

mclase (549)

Entry type

Definition

Classification

msc 20M12

Classification

msc 20M10

Related topic

ReesFactor

Defines

left ideal

Defines

right ideal

Defines

principal ideal

Defines

principal left ideal

Defines

principal right ideal

Generated on Fri Feb 9 19:34:01 2018 by

LaTeXML
Rees factor

Rees factor

Let

I

be an ideal of a

semigroup

S

. Define a

congruence

∼

by

x

∼

y

iff

x

=

y

or

x

,

y

∈

I

.

Then the

Rees factor

of

S

by

I

is the

quotient

S

/

∼

. As a matter of notation, the congruence

∼

is normally suppressed, and the quotient is simply written

S

/

I

.

Note that a Rees factor always has a

zero element

. Intuitively, the quotient identifies all

element

in

I

and the resulting element is a zero element.

Title

Rees factor

Canonical name

ReesFactor

Date of creation

2013-03-22 13:05:46

Last modified on

2013-03-22 13:05:46

Owner

mclase (549)

Last modified by

mclase (549)

Numerical id

4

Author

mclase (549)

Entry type

Definition

Classification

msc 20M12

Classification

msc 20M10

Related topic

Ideal3

Generated on Fri Feb 9 19:34:03 2018 by

LaTeXML