Representing the Characteristics of an algebra
On this page we describe properties that an algebra may have such as:
- commutative
- associative
- distributive
- inverse exists
commutative
An operation is commutative if the order of its operands can be reversed without affecting the result. For example:
x * y = y * x
associative
An operation is associative if the order of doing multiple operations is not important. For example:
x * (y * z) = (x * y) * z
distributive
When there are two operations in this algebra say, + and *, then * is said to be distributive over + if:
x * (y + z) = (x * y) + (x * z)
Algebras
| * distributive over + | Multiplication commutative | Multiplication associative | |
| Real Numbers | yes | yes | yes |
| Complex Numbers | yes | yes | yes |
| Quaternions | yes | no | yes |
| Octonions | yes | no | no |
| Vectors | yes | no | no |
| Matricies | yes | no | yes |
| Multi Vectors | yes | no | no |
For further information see group theory.
Can we determine these properties from the Cayley Table? Well the distributive property does not apply to a single operation so we will investigate the other properties.
Commutative
It is easy to relate the commutative properties of the operation to the Cayley table. Reversing the order of the operands is the same as swapping columns and rows, so an operation is commutative if it is symmetrical about the leading diagonal. Some operations are anti-commutative and we can quickly see this also if reflecting in the leading diagonal changes the sign.
Associative
Inverse exists
