Grassmann Algebra

TheRegressiveProduct.nb 16
Extending this process, we see that the formula remains true for arbitrary Α and Β, providing Γ
m k
p
is simple.
�
�
���Α m � Γ p
����
�
�
����Β
�
� ��� �
�k
p�
����Α
�
� Β � Γ��� � Γ
�m
k p�
p
m � k � p � n � 0
This is an extended version of the Common Factor Axiom. It states that: the regressive product
of two arbitrary elements containing a simple common factor is congruent to that factor.
For applications involving computations in a non-metric space, particularly those with a
geometric interpretation, we will see that the congruence form is not restrictive. Indeed, it will
be quite elucidating. For more general applications in metric spaces we will see that the
associated scalar factor is no longer arbitrary but is determined by the metric imposed.
Special cases of the Common Factor Axiom
In this section we list some special cases of the Common Factor Axiom. We assume, without
explicitly stating, that the common factor is simple.
� Γ p
When the grades do not conform to the requirement shown, the product is zero.
�
�
���Α m � Γ p
����
�
�
����Β
�
� ��� � 0 m�k�p�n�0 �k
p�
If there is no common factor (other than the scalar 1), then the axiom reduces to:
Α m � Β k
� �Α � Β��1 ��
m k
0
m � k � n � 0
The following version yields a sort of associativity for products which are scalar.
�Α � Β��Γ m k p
�Α� m ����Β
�k
�
� ��� ��
p�
0
m � k � p � n � 0
This may be proven from the previous case by noting that each side is equal to ����Α
� Β
�m
k
Dual versions of the Common Factor Axiom
The dual Common Factor Axiom is:
2001 4 5
�
�
���Α m � Γ p
����
�
�
����Β
�
� ��� �
�k
p�
�
�
���Α m � Β k
�
� Γ��� � Γ
p�
p
�� p
m � k � p � 2�n � 0
3.19
3.20
3.21
3.22
�
� ��� � 1.
p�
3.23