Grassmann Algebra

TheRegressiveProduct.nb 11
1 � 1 � � � 1 � Α m �Α m
The dual theorem states the corresponding fact for the regressive product of unit n-elements 1 n .
1 n � 1 n � � � 1 n � Α m �Α m
� Using the GrassmannAlgebra function Dual
The algorithm of the Duality Principle has been encapsulated in the function Dual in
GrassmannAlgebra. Dual takes an expression or list of expressions which comprise an axiom
or theorem and generates the list of dual expressions by transforming them according to the
replacement rules of the Duality Principle.
Example 1
Our first example is to show how Dual may be used to develop dual axioms. For example, to
take the dual of axiom �10 simply enter:
Dual�Α m � Β k
Α m � Β k
Example 2
� ��1� mk  k
� Α m �
�� ��1� ��k�n���m�n� Β � Α
k m
Before applying Dual to axiom �8 however, it is necessary to designate 1 specifically as the
unit of � 0 (that is, 1 0 ), otherwise it will be treated as any other scalar.
Dual��������1 0 ,1 0 �� 0 � : Α m � 1 0 � Α m �
�������1 n ,1 n �� n � : Α m �� 1 n � Α m
Example 3
To apply Dual to an axiom involving more than one statement, collect the statements in a list.
For example, the dual of axiom �9 is obtained as:
�1 �1 �1 Dual���������Α , Α0 ��0 � : 1 � Α�Α , �������Α0 , Α � 0 ����� 0
0 0
0 0 0
�������� 1
���� ,
Αn
1
����
Αn
��� :1�� Α �
n n 1
���� , �������Α, Α � 0 ���� n n n n
Note that in order to simplify the algorithm for Dual, we are purposely frustrating
Mathematica's inbuilt definition of Exists and ForAll by writing them in script.
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