Grassmann Algebra

ExpTheGeneralizedProduct.nb 2
10.9 Properties of the Generalized Product
Summary of properties
10.10 The Triple Generalized Sum Conjecture
� The generalized Grassmann product is not associative
� The triple generalized sum
The triple generalized sum conjecture
� Exploring the triple generalized sum conjecture
� An algorithm to test the conjecture
10.11 Exploring Conjectures
A conjecture
� Exploring the conjecture
10.12 The Generalized Product of Intersecting Elements
� The case Λ < p
� The case Λ ≥ p
� The special case of Λ = p
10.13 The Generalized Product of Orthogonal Elements
� The generalized product of totally orthogonal elements
� The generalized product of partially orthogonal elements
10.14 The Generalized Product of Intersecting Orthogonal Elements
The case Λ < p
� The case Λ ≥ p
10.15 Generalized Products in Lower Dimensional Spaces
Generalized products in 0, 1, and 2-spaces
� 0-space
� 1-space
� 2-space
10.16 Generalized Products in 3-Space
To be completed
10.1 Introduction
In this chapter we define and explore a new product which we call the generalized Grassmann
product. The generalized Grassmann product was originally developed by the author in order to
treat the Clifford product of general elements in a succinct manner. The Clifford product of
general elements can lead to quite complex expressions, however we will show that such
expressions always reduce to simple sums of generalized Grassmann products. We discuss the
Clifford product in Chapter 12.
In its own right, the generalized Grassmann product leads to a suite of useful identities between
expressions involving exterior and interior products, and also has some useful geometric
applications.
We have chosen to call it the generalized Grassmann product because, in specific cases, it
reduces to either an exterior or an interior product.
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