Grassmann Algebra

TheComplement.nb 5
This axiom is of central importance in the development of the properties of the complement and
interior, inner and scalar products, and formulae relating these with exterior and regressive
products. In particular, it permits us to be able to consistently generate the complements of basis
m-elements from the complements of basis 1-elements, and hence via the linearity axiom, the
complements of arbitrary elements.
We may verify that these are dual formulae by applying the GrassmannAlgebra Dual function
to either of them. For example:
��������� � �
Dual�Α � Β �Αm � Βk �
m k
���������
Α � Β ��
� �
Αm � Βk
m k
The forms 5.3 and 5.4 may be written for any number of elements. To see this, let Β �Γ� ∆
k p q
and substitute for Β in expression 5.3:
k
��������������
� ∆ �
����� �����������
Αm � Γp � ∆ �
����� ����� �����
Αm � Γp � ∆q
q
q
Α m � Γ p
In general then, expressions 5.3 and 5.4 may be stated in the equivalent forms:
����������������������
Α � Β � � � Γ �
����� �����
Αm � Βk � � �
�����
Γp
m k p
����������������������
Α � Β � � � Γ �
����� �����
Αm � Βk � � �
�����
Γp
m k p
In Grassmann's work, this axiom was hidden in his notation. However, since modern notation
explicitly distinguishes the progressive and regressive products, this axiom needs to be
explicitly stated.
The complement of a complement axiom
� 4
����� : The complement of the complement of an element is equal (apart from a possible
sign) to the element itself.
����� ����� Φ
Α � ��1� �Αm
m
Axiom 1
����� says that the complement of an m-element is an (nÐm)-element. Clearly then the
complement of an (nÐm)-element is an m-element. Thus the complement of the complement of
an m-element is itself an m-element.
In the interests of symmetry and simplicity we will require that the complement of the
complement of an element is equal (apart from a possible sign) to the element itself. Although
consistent algebras could no doubt be developed by rejecting this axiom, it will turn out to be an
2001 4 5
5.5
5.6
5.7