ternary ring

ternary ring

Let

R

be a set containing at least two distinct elements

0

and

1

, and

*

a ternary operation on

R

. Write

a

*

b

*

c

the image of

(

a

,

b

,

c

)

under

*

. We call

(

R

,

*

,

0

,

1

)

, or simply just

R

, a

ternary ring

if

1.

a

*

0

*

b

=

0

*

a

*

b

=

b

for any

a

,

b

∈

R

,

2.

1

*

a

*

0

=

a

*

1

*

0

=

a

for any

a

∈

R

,

3.

given

a

,

b

,

c

,

d

∈

R

with

a

≠

c

, the equation

x

*

a

*

b

=

x

*

c

*

d

has a unique solution for

x

,

4.

given

a

,

b

,

c

∈

R

, the equation

a

*

b

*

x

=

c

has a unique solution for

x

,

5.

given

a

,

b

,

c

,

d

∈

R

, with

a

≠

c

, the system of equations

{

a

*

x

*

y

=

b

c

*

x

*

y

=

d

has a unique solution for

(

x

,

y

)

.

Given a ternary ring

R

, we may form two binary operations on

R

, one called the

addition

+

, and the other the

multiplication

⋅

on

R

:

a

+

b

:=

a

*

1

*

b

,

a

⋅

b

:=

a

*

b

*

0

.

Proposition 1

.

(

R

,

+

,

0

)

and

(

R

-

{

0

}

,

⋅

,

1

)

are loops, and

0

is the

zero element

in

R

under the multiplication

⋅

.

Proof.

We first show that

(

R

,

+

,

0

)

is a loop. Given

a

,

b

∈

R

, there is a unique

c

∈

R

such that

a

*

1

*

c

=

b

, but this is exactly

a

+

c

=

b

. In addition, there is a unique

d

∈

R

such that

d

*

1

*

a

=

b

=

d

*

0

*

b

. But

d

*

1

*

a

=

b

means

d

+

a

=

b

. This shows that

(

R

,

+

)

is a quasigroup. Now,

a

+

0

=

a

*

1

*

0

=

a

and

0

+

a

=

0

*

1

*

a

=

a

, so

0

is the identity with respect to

+

. Therefore,

(

R

,

+

,

0

)

is a loop.

Next we show that

(

S

,

⋅

,

1

)

is a loop, where

S

=

R

-

{

0

}

. Given

a

,

b

∈

S

, there is a unique

c

∈

R

such that

c

*

a

*

0

=

b

=

c

*

0

*

b

=

b

, since

a

≠

0

. From

c

*

a

*

0

=

b

, we get

c

⋅

a

=

b

. Furthermore,

c

≠

0

, for otherwise

b

=

c

*

a

*

0

=

0

*

a

*

0

=

0

, contradicting the fact that

b

∈

S

. In addition, there is a unique

d

∈

R

such that we have a system of equations

a

*

d

*

0

=

b

and

0

*

d

*

0

=

0

. From

a

*

d

*

0

=

b

we get

a

⋅

d

=

b

. Furthermore

d

≠

0

, for otherwise

b

=

a

*

d

*

0

=

a

*

0

*

0

=

0

, contradicting the fact that

b

∈

S

. Thus,

(

S

,

⋅

,

1

)

is a quasigroup. Now,

a

⋅

1

=

a

*

1

*

0

=

a

and

1

⋅

a

=

1

*

a

*

0

=

a

, showing that

(

S

,

⋅

,

1

)

is a loop.

Finally, for any

a

∈

R

,

a

⋅

0

=

a

*

0

*

0

=

0

and

0

⋅

a

=

0

*

a

*

0

=

0

.
∎

Another property of a ternary ring is that, if the ternary ring is finite, then conditions 4 and 5 are equivalent in the presence of the first three.

Let

R

be a ternary ring, and

a

,

b

,

c

are arbitrary elements of

R

.

R

is said to be

linear

if

a

*

b

*

c

=

a

⋅

b

+

c

for all

a

,

b

,

c

∈

R

,

left distributive

if

a

⋅

(

b

+

c

)

=

a

⋅

b

+

a

⋅

c

,

right distributive

if

(

a

+

b

)

⋅

c

=

a

⋅

c

+

b

⋅

c

, and

distributive

if it is both left and right distributive.

For example, any

division ring

D

, associative or not, is a linear ternary ring if we define the ternary operation

*

on

D

by

a

*

b

*

c

:=

a

⋅

b

+

c

. Any associative division ring

D

is a distributive ternary ring. This easy verification is left to the reader.

Semifields

, near fields are also examples of ternary rings.

Remark

. Ternary rings were invented by Marshall Hall in his studies of axiomatic projective and

affine planes

. Therefore, a ternary ring is also called a

Hall ternary ring

, or a

planar ternary ring

. It can be shown that in every affine plane, one can set up a

coordinate system

, and from this coordinate system, one can construct a ternary ring. Conversely, given any ternary ring, one can define an affine plane so that its coordinate system corresponds to this ternary ring.

References

1

R. Artzy,

Linear

Geometry

, Addison-Wesley (1965)

Title

ternary ring

Canonical name

TernaryRing

Date of creation

2013-03-22 18:30:52

Last modified on

2013-03-22 18:30:52

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

6

Author

CWoo (3771)

Entry type

Definition

Classification

msc 51A35

Classification

msc 51E15

Classification

msc 51A25

Synonym

planar ternary ring

Synonym

Hall ternary ring

Defines

linear ternary ring

Defines

left distributive

Defines

right distributive

Defines

distributive

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