Grassmann Algebra

TheComplement.nb 6
essential underpinning to the development of the standard metric concepts to which we are
accustomed. For example, its satisfaction will require the metric tensor to be symmetric.
It will turn out that, again for compliance with standard results, the index Φ is m(nÐm), but this
result is more in the nature of a theorem than an axiom.
The complement of unity
� 5
����� : The complement of unity is equal to the unit n-element.
����
1 � 1n � � �e1 � e2 � � � en
This is the axiom which finally enables us to define the unit n-element. This axiom requires that
in a metric space the unit n-element be identical to the complement of unity. The hitherto
unspecified scalar constant � may now be determined from the specific complement mapping
or metric under consideration.
The dual of this axiom is:
����
Dual� 1 � 1n �
0
�
1 �� 10
n
Hence taking the complement of expression 5.8 and using this dual axiom tells us that the
complement of the complement of unity is unity. This result clearly complies with axiom 4
����� .
���� ���� ����
1 � 1n � 1
We can now write the unit n-element as 1
���� instead of 1 in any Grassmann algebras that have a
n
metric. For example in a metric space, 1
���� becomes the unit for the regressive product.
2001 4 5
Α � 1
m ���� � Α
m
5.8
5.9
5.10