Grassmann Algebra

TheRegressiveProduct.nb 4
The grade of the space to which the regressive product belongs is nÐ(m+k) = nÐ((nÐm')+(nÐk'))
= (m'+k')Ðn.
Finally, since the primes are no longer necessary we drop them. Then the final form of the
axiom dual to �6, which we label �6, becomes:
�Α m �� m , Β k
�� k � � Α m � Β k
� �
m�k�n
In words, this says that the regressive product of an m-element and a k-element is an (m+kÐn)element.
The dual of axiom �8
Axiom �8 says:
There is a unit scalar which acts as an identity under the exterior product.
�������1, 1 �� 0 � : Α m � 1 � Α m
For simplicity we do not normally display designated scalars with an underscripted zero.
However, the duality transformation will be clearer if we rewrite the axiom with 1 0 in place of 1.
�������1 0 ,1 0 �� 0 � : Α m � 1 0 � Α m
Replace � with �, and the grades of elements and spaces by their complementary grades.
�������1 n ,1 n �� n � : Α
n�m � 1 n � Α
n�m
Replace arbitrary grades m with nÐm' (n is not an arbitrary grade).
Drop the primes.
�������1 n ,1 n �� n � : Α m' � 1 n � Α m'
�������1 n ,1 n �� n � : Α m � 1 n � Α m
In words, this says that there is a unit n-element which acts as an identity under the regressive
product.
The dual of axiom �10
Axiom �10 says:
The exterior product of elements of odd grade is anti-commutative.
Α m � Β k
� ��1� mk  k
� Α m
Replace � with �, and the grades of elements by their complementary grades.
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