Grassmann Algebra

TheInteriorProduct.nb 36
Α m � Β k
Α m � Β k
� ��1� �m�k���n��m�k�� �1 � �Α m �Β k
� ��1� m��n�m��k��n�k� ��1 �Α m � � �1 �Β k
Α m ���� Β k
� ��1� m��n�m� ��1 �Α m � �Β k
�
�
6.92
These formulae show that any result in the Grassmann algebra involving exterior, regressive or
interior products, or complements, could be expressed in terms of the generalized cross product
alone. This is somewhat reminiscent of the role played by the Scheffer stroke (or Pierce arrow)
symbol of Boolean algebra.
Cross product formulae
The complement of a cross product
The complement of a cross product of two elements is, but for a possible sign, the exterior
product of the elements.
��������� �m�k���n��m�k�� Α �Β � ��1� �Αm � Β
m k
k
The Common Factor Axiom for cross products
The Common Factor Axiom can be written for m+k+p = n.
�
�
���Α m �Γ p
����
�
�
����Β
�
���� �
�k
p�
����Γ
�p
� �Α m � Β k
Product formulae for cross products
� �����Γ
� p
� ��1�p��n�p� � ����Γ
���� �Α �Β
�p
m k
� �����Γ
� p
Of course, many of the product formulae derived previously have counterparts involving cross
products. For example the complement of formula 3.41 may be written:
2001 4 5
�Α �Β��x � ��1�
m k
n�1 ���Α ���� x� �Β� ��1�
m k
m �Α � �Β ���� x��
m k
6.93
6.94
6.95
6.96
6.97