Grassmann Algebra

ExpTheGeneralizedProduct.nb 23
10.9 Properties of the Generalized Product
Summary of properties
The following simple results may be obtained from the definition and results derived above.
� The generalized product is distributive with respect to addition
�Α �Β������ �Γ � Α����� �Γ
m k Λ p m Λ p
�Β����� �Γ
k Λ p
Γ����� ��Α �Β� � Γ����� �Α �Γ����� �Β
p Λ m k p Λ m p Λ k
� The generalized product with a zero element is zero
0����� Λ �Α m � Α m ����� Λ �0 � 0
� The generalized product with a scalar reduces to the field product
The only non-zero generalized product with a scalar is of order zero, leading to the ordinary
field product.
a����� 0 �Α m � Α m ����� 0 �a � a � Α m �Α m ���� a � a Α m
a����� 0 �b � a � b � a ���� b � ab
� The generalized products of a scalar with an m-element are zero
For Λ greater than zero
2001 4 26
a����� Λ �Α m �Α m ����� Λ �a � 0 Λ�0
10.19
10.20
10.21
10.22
10.23
10.24