hypergroup

hypergroup

Hypergroups

are

generalizations

of groups. Recall that a group is set with a

binary operation

on it satisfying a number of conditions. If this binary operation is taken to be

multivalued

, then we arrive at a hypergroup. In order to make this precise, we need some preliminary

concepts

:

Definition

. A

hypergroupoid

, or

multigroupoid

, is a non-empty set

G

, together with a multivalued function

⋅

:

G

×

G

⇒

G

called the

multiplication

on

G

.

We write

a

⋅

b

, or simply

a

⁢

b

, instead of

⋅

(

a

,

b

)

. Furthermore, if

a

⁢

b

=

{

c

}

, then we use the abbreviation

a

⁢

b

=

c

.

A hypergroupoid is said to be

commutative

if

a

⁢

b

=

b

⁢

a

for all

a

,

b

∈

G

. Defining

associativity

of

⋅

on

G

, however, is trickier:

Given a hypergroupoid

G

, the multiplication

⋅

induces a binary operation (also written

⋅

) on

P

⁢

(

G

)

, the

powerset

of

P

, given by

A

⋅

B

:=

⋃

{

a

⋅

b

∣

a

∈

A

⁢

and

⁢

b

∈

B

}

.

As a result, we have an induced

groupoid

P

⁢

(

G

)

. Instead of writing

{

a

}

⁢

B

,

A

⁢

{

b

}

, and

{

a

}

⁢

{

b

}

, we write

a

⁢

B

,

A

⁢

b

, and

a

⁢

b

instead. From now on, when we write

(

a

⁢

b

)

⁢

c

, we mean
“first, take the

product

of

a

and

b

via the multivalued binary operation

⋅

on

G

, then take the product of the set

a

⁢

b

with the

element

c

, under the induced binary operation on

P

⁢

(

G

)

”. Given a hypergroupoid

G

, there are two types of associativity we may define:

Type 1.

(

a

⁢

b

)

⁢

c

⊆

a

⁢

(

b

⁢

c

)

, and

Type 2.

a

⁢

(

b

⁢

c

)

⊆

(

a

⁢

b

)

⁢

c

.

G

is said to be

associative

if it

satisfies

both types of associativity laws. An associative hypergroupoid is called a

hypersemigroup

. We are now ready to formally define a hypergroup.

Definition

. A

hypergroup

is a hypersemigroup

G

such that

a

⁢

G

=

G

⁢

a

=

G

for all

a

∈

G

.

For example, let

G

be a group and

H

a

subgroup

of

G

. Let

M

be the

collection

of all

left cosets

of

H

in

G

. For

a

⁢

H

,

b

⁢

H

∈

M

, set

a

⁢

H

⋅

b

⁢

H

:=

{

c

⁢

H

∣

c

=

a

⁢

h

⁢

b

⁢

,

⁢

h

∈

H

}

.

Then

M

is a hypergroup with multiplication

⋅

.

If the multiplication in a hypergroup

G

is single-valued, then

G

is a

2

-group (

http://planetmath.org/PolyadicSemigroup

), and therefore a group (see proof here (

http://planetmath.org/PolyadicSemigroup

)).

Remark

. A hypergroup is also known as a

multigroup

, although some call a multigroup as a hypergroup with a designated

identity element

e

, as well as a designated

inverse

for every element with respect

e

. Actually

identities

and inverses may be defined more generally for hypergroupoids:

Let

G

be a hypergroupoid. Identity elements are defined via the following three sets:

1.

(set of

left identities

):

I

L

⁢

(

G

)

:=

{

e

∈

G

∣

a

∈

e

⁢

a

⁢

for all

⁢

a

∈

G

}

,

2.

(set of

right identities

):

I

R

⁢

(

G

)

:=

{

e

∈

G

∣

a

∈

a

⁢

e

⁢

for all

⁢

a

∈

G

}

, and

3.

(set of

identities

):

I

⁢

(

G

)

=

I

L

⁢

(

G

)

∩

I

R

⁢

(

G

)

.

e

∈

L

⁢

(

G

)

is called an

absolute identity

if

e

⁢

a

=

a

⁢

e

=

a

for all

a

∈

G

. If

e

,

f

∈

G

are both absolute identities, then

e

=

e

⁢

f

=

f

, so

G

can have at most one absolute identity.

Suppose

e

∈

I

L

⁢

(

G

)

∪

I

R

⁢

(

G

)

and

a

∈

G

. An element

b

∈

G

is said to be a

left inverse

of

a

with respect to

e

if

e

∈

b

⁢

a

.

Right inverses

of

a

are defined similarly. If

b

is both a left and a right inverse of

a

with respect to

e

, then

b

is called an

inverse

of

a

with respect to

e

.

Thus, one may say that a multigroup is a hypergroup

G

with an identity

e

∈

G

, and a

function

:

-

1

G

→

G

such that

a

-

1

:=

-

1

(

a

)

is an inverse of

a

with respect to

e

.

In the example above,

M

is a multigroup in the sense given in the remark above. The designated identity is

H

(in fact, this is the only identity in

M

), and for every

a

⁢

H

∈

M

, its designated inverse is provided by

a

-

1

⁢

H

(of course, this may not be its only inverse, as any

b

⁢

H

such that

a

⁢

h

⁢

b

=

e

for some

h

∈

H

will do).

References

1

R. H. Bruck,

A Survey on Binary Systems

, Springer-Verlag, New York, 1966.

2

M. Dresher, O. Ore,

Theory of Multigroups

, Amer. J. Math. vol. 60, pp. 705-733, 1938.

3

J.E. Eaton, O. Ore,

Remarks on Multigroups

, Amer. J. Math. vol. 62, pp. 67-71, 1940.

4

L. W. Griffiths,

On Hypergroups, Multigroups, and Product Systems

, Amer. J. Math. vol. 60, pp. 345-354, 1938.

5

A. P. Dičman,

On Multigroups whose Elements are Subsets of a Group

, Moskov. Gos. Ped. Inst. Uč. Zap. vol. 71, pp. 71-79, 1953

Title

hypergroup

Canonical name

Hypergroup

Date of creation

2013-03-22 18:38:22

Last modified on

2013-03-22 18:38:22

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

9

Author

CWoo (3771)

Entry type

Definition

Classification

msc 20N20

Synonym

multigroupoid

Synonym

multisemigroup

Synonym

multigroup

Related topic

group

Defines

hypergroupoid

Defines

hypersemigroup

Defines

left identity

Defines

right identity

Defines

identity

Defines

absolute identity

Defines

left inverse

Defines

right inverse

Defines

inverse

Defines

absolute identity

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