Grassmann Algebra

TheRegressiveProduct.nb 5
Α � Β
n�m n�k
� ��1�mk Β � Α
n�k n�m
Replace arbitrary grades m with nÐm', k with nÐk'.
Α
m' � Β k'
Drop the primes.
Α m � Β k
� ��1� �n�m'���n� k'� Β � Α
k' m'
� ��1� �n�m���n� k�  k
� Α m
In words this says that the regressive product of elements of odd complementary grade is anticommutative.
Summary: The duality transformation algorithm
The algorithm for the duality transformation may be summarized as follows:
1. Replace � with �, and the grades of elements and spaces by their complementary grades.
2. Replace arbitrary grades m with nÐm', k with nÐk'. Drop the primes.
3.3 Properties of the Regressive Product
Axioms for the regressive product
In this section we collect the results of applying the duality algorithm above to the exterior
product axioms �6 to �12. Axioms �1 to �5 transform unchanged since there are no
products involved.
� �6: The regressive product of an m-element and a k-element is an (m+kÐn)-element.
�Α m �� m , Β k
� �7: The regressive product is associative.
2001 4 5
�Α m � Β k
�� k � � Α m � Β k
��à r
� Α m ��Β k
� �
m�k�n
� Γ �
r
3.2
3.3