Grassmann Algebra

TheRegressiveProduct.nb 15
It is easy to see that by using the anti-commutativity axiom �10, that the axiom may be
arranged in any of a number of alternative forms, the most useful of which are:
�
�
���Α m � Γ p
����Γ
�p
����
�
�
����Γ
�
� ��� �
�p
k�
�
�
�
� Α��� �
m�
����Γ
�
� ��� �
�p
k�
����Γ
�p
���Α m � Γ p
�
� Β��� � Γ
k�
p
� Α m � Β k
����
� Γ
� p
m � k � p � n
m � k � p � n
Historical Note
The approach we have adopted in this chapter of treating the common factor relation as an
axiom is effectively the same as Grassmann used in his first Ausdehnungslehre (1844) but
differs from the approach that he used in his second Ausdehnungslehre (1862). (See Chapter
3, Section 5 in Kannenberg.) In the 1862 version Grassmann proves this relation from
another which is (almost) the same as the Complement Axiom that we introduce in Chapter
5: The Complement. Whitehead [ ], and other writers in the Grassmannian tradition follow
his 1862 approach.
The relation which Grassmann used in the 1862 Ausdehnungslehre is in effect equivalent to
assuming the space has a Euclidean metric (his ErgŠnzung or supplement). However the
Common Factor Axiom does not depend on the space having a metric; that is, it is
completely independent of any correspondence we set up between � m and �
n�m . Hence we
would rather not adopt an approach which introduces an unnecessary constraint, especially
since we want to show later that his work is easily extended to more general metrics than the
Euclidean.
Extension of the Common Factor Axiom to general elements
The axiom has been stated for simple elements. In this section we show that it remains valid for
general (possibly non-simple) elements, provided that the common factor remains simple.
Consider two simple elements Α1
m
����Α1
� m
����Α2
� m
�
� ��� �
p�
����Β
�
� ��� �
�k
p�
����Α1
�
� Β � Γ��� � Γ
� m k p�
p
�
� ��� �
p�
����Β
�
� ��� �
�k
p�
����Α2
�
� Β � Γ��� � Γ
� m k p�
p
and Α2 . Then the Common Factor Axiom can be written:
m
Adding these two equations and using the distributivity of � and � gives:
2001 4 5
�����Α1
� m
�Α2
m
�
����� �
p�
����Β
�
� ��� �
�k
p�
�����Α1
� m
�Α2��Β
m k
�
� Γ��� � Γ
p�
p
3.17
3.18