Let be a set containing at least two distinct elements and , and a ternary operation on . Write the image of under . We call , or simply just , a ternary ring if
for any ,
for any ,
given with , the equation has a unique solution for ,
given , the equation has a unique solution for ,
given , with , the system of equations
has a unique solution for .
and are loops, and is the zero element in under the multiplication .
We first show that is a loop. Given , there is a unique such that , but this is exactly . In addition, there is a unique such that . But means . This shows that is a quasigroup. Now, and , so is the identity with respect to . Therefore, is a loop.
Next we show that is a loop, where . Given , there is a unique such that , since . From , we get . Furthermore, , for otherwise , contradicting the fact that . In addition, there is a unique such that we have a system of equations and . From we get . Furthermore , for otherwise , contradicting the fact that . Thus, is a quasigroup. Now, and , showing that is a loop.
Finally, for any , and . ∎
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 be a ternary ring, and are arbitrary elements of . is said to be linear if for all , left distributive if , right distributive if , and distributive if it is both left and right distributive.
For example, any division ring , associative or not, is a linear ternary ring if we define the ternary operation on by . Any associative division ring 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.
- 1 R. Artzy, Linear Geometry, Addison-Wesley (1965)
|Date of creation||2013-03-22 18:30:52|
|Last modified on||2013-03-22 18:30:52|
|Last modified by||CWoo (3771)|
|Synonym||planar ternary ring|
|Synonym||Hall ternary ring|
|Defines||linear ternary ring|