examples of semigroups

examples of semigroups

Examples of semigroups

are numerous. This entry presents some of the most common examples.

1.

The set

ℤ

of

integers

with

multiplication

is a

semigroup

, along with many of its subsets (

subsemigroups

):

(a)

The set of non-negative integers

(b)

The set of positive integers

(c)

n

⁢

ℤ

, the set of all

integral

multiples

of an integer

n

(d)

For any prime

p

, the set of

{

p

i

∣

i

≥

n

}

, where

n

is a non-negative integer

(e)

The set of all

composite

integers

2.

ℤ

n

, the set of all integers modulo an integer

n

, with integer multiplication modulo

n

. Here, we may find examples of

nilpotent

and

idempotent elements

,

relative inverses

, and

eventually periodic

elements:

(a)

If

n

=

p

m

, where

p

is prime, then every non-zero element containing a factor of

p

is nilpotent. For example, if

n

=

16

, then

6

4

=

0

.

(b)

If

n

=

2

⁢

p

, where

p

is an

odd prime

, then

p

is a non-trivial idempotent element (

p

2

=

p

), and since

2

p

-

1

≡

1

(

mod

p

)

by Fermat’s little theorem, we see that

a

=

2

p

-

2

is a relative inverse of

2

, as

2

⋅

a

⋅

2

=

2

and

a

⋅

2

⋅

a

=

a

(c)

If

n

=

2

m

⁢

p

, where

p

is an odd prime, and

m

>

1

, then

2

is eventually periodic. For example,

n

=

96

, then

2

2

=

4

,

2

3

=

8

,

2

4

=

16

,

2

5

=

32

,

2

6

=

64

,

2

7

=

32

,

2

8

=

64

, etc…

3.

The set

M

n

⁢

(

R

)

of

n

×

n

square matrices

over a ring

R

, with

matrix multiplication

, is a semigroup. Unlike the previous two examples,

M

n

⁢

(

R

)

is not

commutative

.

4.

The set

E

⁢

(

A

)

of functions on a set

A

, with functional composition, is a semigroup.

5.

Every group is a semigroup, as well as every monoid.

6.

If

R

is a ring, then

R

with the

ring multiplication

(ignoring

addition

) is a semigroup (with

0

).

7.

Group with Zero

. A semigroup

S

is called a

group with zero

if it contains a

zero element

0

, and

S

-

{

0

}

is a

subgroup

of

S

. In

R

in the previous example is a

division ring

, then

R

with the ring multiplication is a group with zero. If

G

is a group, by adjoining

G

with an extra symbol

0

, and extending the domain of group multiplication

⋅

by defining

0

⋅

a

=

a

⋅

0

=

0

⋅

0

:=

0

for all

a

∈

G

, we get a group with zero

S

=

G

∪

{

0

}

.

8.

As mentioned earlier, every monoid is a semigroup. If

S

is not a monoid, then it can be embedded in one: adjoin a symbol

1

to

S

, and extend the semigroup multiplication

⋅

on

S

by defining

1

⋅

a

=

a

⋅

1

=

a

and

1

⋅

1

=

1

, we get a monoid

M

=

S

∪

{

1

}

with multiplicative identity

1

. If

S

is already a monoid with

identity

1

, then adjoining

1

′

to

S

and repeating the remaining step above gives us a new monoid with identity

1

′

. However,

1

is no longer an identity, as

1

′

=

1

⋅

1

′

.

Title

examples of semigroups

Canonical name

ExamplesOfSemigroups

Date of creation

2013-03-22 18:37:16

Last modified on

2013-03-22 18:37:16

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

7

Author

CWoo (3771)

Entry type

Example

Classification

msc 20M99

Synonym

group with 0

Defines

group with zero

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