polyadic semigroup

polyadic semigroup

Recall that a

semigroup

is a non-empty set, together with an

associative

binary operation

on it.

Polyadic semigroups

are

generalizations

of semigroups, in that the associative binary operation is replaced by an associative

n

-ary

operation

. More precisely, we have

Definition

. Let

n

be a positive integer at least

2

. A

n

-semigroup

is a non-empty set

S

, together with an

n

-ary operation

f

on

S

, such that

f

is associative:

f

⁢

(

f

⁢

(

a

1

,

…

,

a

n

)

,

a

n

+

1

,

…

,

a

2

⁢

n

-

1

)

=

f

⁢

(

a

1

,

…

,

f

⁢

(

a

i

,

…

,

a

i

+

n

-

1

)

,

…

,

f

2

⁢

n

-

1

)

for every

i

∈

{

1

,

…

,

n

}

. A

polyadic semigroup

is an

n

-semigroup for some

n

.

An

n

-semigroup

S

(with the associated

n

-ary operation

f

) is said to be

commutative

if

f

is commutative. An

element

e

∈

S

is said to be an

identity element

, or an

f

-

identity

, if

f

⁢

(

a

,

e

,

…

,

e

)

=

f

⁢

(

e

,

a

,

…

,

e

)

=

⋯

=

f

⁢

(

e

,

e

,

…

,

a

)

=

a

for all

a

∈

S

. If

S

is commutative, then

e

is an identity in

S

if

f

⁢

(

a

,

e

,

…

,

e

)

=

a

.

Every semigroup

S

has an

n

-semigroup

structure

: define

f

:

S

n

→

S

by

f

⁢

(

a

1

,

a

n

⁢

…

,

a

n

)

=

a

1

⋅

a

2

⁢

⋯

⋅

a

n

(1)

The associativity of

f

is induced from the associativity of

⋅

.

Definition

. An

n

-semigroup

S

is called an

n

-group

if, in the

equation

f

⁢

(

x

1

,

…

,

x

n

)

=

a

,

(2)

any

n

-

1

of the

n

variables

x

i

are replaced by elements of

G

, then the equation with the remaining one variable has

at least one

solution in that variable. A

polyadic group

is just an

n

-group for some integer

n

.

n

-groups are generalizations of groups. Indeed, a

2

-group is just a group.

Proof.

Let

G

be a

2

-group. For

a

,

b

∈

G

, we write

a

⁢

b

instead of

f

⁢

(

a

,

b

)

. Given

a

∈

G

, there are

e

1

,

e

2

∈

G

such that

a

⁢

e

1

=

a

and

e

2

⁢

a

=

a

. In

addition

, there are

x

,

y

∈

G

such that

x

⁢

a

=

e

2

and

a

⁢

y

=

e

1

. So

e

2

=

x

⁢

a

=

x

⁢

(

a

⁢

e

1

)

=

(

x

⁢

a

)

⁢

e

1

=

e

2

⁢

e

1

=

e

2

⁢

(

a

⁢

y

)

=

(

e

2

⁢

a

)

⁢

y

=

a

⁢

y

=

e

1

.

Next, suppose

a

⁢

e

1

=

a

⁢

e

3

=

a

. Then the equation

e

2

⁢

a

=

a

from the previous paragraph as well as the subsequent discussion shows that

e

1

=

e

2

=

e

3

. This means that, for every

a

∈

G

, there is a unique

e

a

∈

G

such that

e

a

⁢

a

=

a

⁢

e

a

=

a

. Since

e

a

2

⁢

a

=

e

a

⁢

(

e

a

⁢

a

)

=

e

a

⁢

a

=

a

=

a

⁢

e

a

=

(

a

⁢

e

a

)

⁢

e

a

=

a

⁢

e

a

2

, we see that

e

a

is

idempotent

:

e

a

2

=

e

a

.

Now, pick any

b

∈

G

. Then there is

c

∈

G

such that

b

=

c

⁢

e

a

. So

b

⁢

e

a

=

(

c

⁢

e

a

)

⁢

e

a

=

c

⁢

e

a

2

=

c

⁢

e

a

=

b

. From the last two paragraphs, we see that

e

a

=

e

b

. This shows that there is a

e

∈

G

such that

a

⁢

e

=

e

⁢

a

=

a

for all

a

∈

G

. In other words,

e

is the identity with respect to the binary operation

f

.

Finally, given

a

∈

G

, there are

b

,

c

∈

G

such that

a

⁢

b

=

c

⁢

a

=

e

. Then

c

=

c

⁢

e

=

c

⁢

(

a

⁢

b

)

=

(

c

⁢

a

)

⁢

b

=

e

⁢

b

=

b

. In addition, if

a

⁢

b

1

=

a

⁢

b

2

=

e

, then, from the equation

c

⁢

a

=

e

, we get

b

1

=

c

=

b

2

. This shows

b

is the unique

inverse

of

a

with respect to binary operation

f

. Hence,

G

is a group.
∎

Every group has a structure of an

n

-group, where the

n

-ary operation

f

on

G

is defined by the equation (1) above. Interestingly, Post has proved that, for every

n

-group

G

, there is a group

H

, and an

injective function

ϕ

:

G

→

H

with the following

properties

:

1.

ϕ

⁢

(

G

)

generates

H

2.

ϕ

⁢

(

f

⁢

(

a

1

,

…

,

a

n

)

)

=

ϕ

⁢

(

a

1

)

⁢

⋯

⁢

ϕ

⁢

(

a

n

)

If we call the group

H

with the two above properties a

covering group

of

G

, then Post’s

theorem

states that every

n

-group has a covering group.

From Post’s result, one has the following corollary: an

n

-semigroup

G

is an

n

-group iff equation (2) above has

exactly one

solution in the remaining variable, when

n

-

1

of the

n

variables are replaced by elements of

G

.

References

HB

R. H. Bruck,

A Survey of Binary Systems

, Springer-Verlag, 1966

EP

E. L. Post,

Polyadic groups

, Trans. Amer. Math. Soc., 48, 208-350, 1940, MR 2, 128

WD

W. Dörnte,

Untersuchungen über einen verallgemeinerten Gruppenbegriff

, Math. Z. 29, 1-19, 1928

Title

polyadic semigroup

Canonical

name

PolyadicSemigroup

Date of creation

2013-03-22 18:37:47

Last modified on

2013-03-22 18:37:47

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

10

Author

CWoo (3771)

Entry type

Definition

Classification

msc 20N15

Classification

msc 20M99

Synonym

n-semigroup

Synonym

n-group

Defines

n

-semigroup

Defines

n

-group

Defines

polyadic group

Defines

covering group

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