Let $$ n $$ be a non-zero natural number in $$ \mathbb{N} $$and let $$ V $$ be a $$ n $$-dimensional real or complex vector space. I wish to work in the Grassmann-Cayley algebra $$ \left(\bigwedge\left(V\right),+,\cdot,\langle\cdot,\cdot\rangle,\wedge,\vee,\star\right) $$ of $$ V $$. Let $$ \mathbf{A} $$ and $$ \mathbf{B} $$ be arbitrary $$ k $$- and $$ m $$-vectors in $$ \bigwedge(V) $$, respectively. Give an explicit formula definition of the regressive product $$ \mathbf{A} \vee \mathbf{B} $$ of $$ \mathbf{A} $$ and $$ \mathbf{B} $$.

The regressive product, also known as the meet or the Hodge dual product, is a fundamental operation in the Grassmann-Cayley algebra, and it is defined as follows:

Given two multivectors $ \mathbf{A} $ and $ \mathbf{B} $ in $ \bigwedge(V) $, where $ \mathbf{A} $ is a $ k $-vector and $ \mathbf{B} $ is an $ m $-vector, the regressive product $ \mathbf{A} \vee \mathbf{B} $ is defined as a $(k + m - n)$-vector by the formula:

\[
\mathbf{A} \vee \mathbf{B} = \left(\mathbf{A} \wedge \mathbf{B}^\star\right)^\star
\]

Here:

- $ \mathbf{A} \wedge \mathbf{B}^\star $ is the exterior (wedge) product of $ \mathbf{A} $ and the Hodge dual $ \mathbf{B}^\star $ of $ \mathbf{B} $.
- $ \mathbf{B}^\star $ is the Hodge dual of $ \mathbf{B} $ in $ \bigwedge(V) $.
- The Hodge dual $ \star $ is an operation that maps a $ m $-vector to an $(n-m)$-vector, where $ n $ is the dimension of the vector space $ V $.
- $ \left(\mathbf{A} \wedge \mathbf{B}^\star\right)^\star $ is the Hodge dual of the exterior product, which results in a multivector of degree $ k + m - n $.

The regressive product $ \vee $ has the following key properties:

1. **Antisymmetry:** $ \mathbf{A} \vee \mathbf{B} = (-1)^{km}\mathbf{B} \vee \mathbf{A} $.
2. **Associativity:** $ (\mathbf{A} \vee \mathbf{B}) \vee \mathbf{C} = \mathbf{A} \vee (\mathbf{B} \vee \mathbf{C}) $.
3. **Distributivity:** $ \mathbf{A} \vee (\mathbf{B} + \mathbf{C}) = \mathbf{A} \vee \mathbf{B} + \mathbf{A} \vee \mathbf{C} $.

Thus, the regressive product in the Grassmann-Cayley algebra generalizes the concept of intersection (or meet) in projective geometry and plays a crucial role in multilinear algebra and geometric algebra.

Would you like to explore this concept further, or do you have specific questions about its properties or applications?

### Further Questions:
1. How does the regressive product relate to the exterior product geometrically?
2. Can you provide an example calculation of the regressive product in $ \mathbb{R}^3 $?
3. What are the implications of the regressive product in projective geometry?
4. How is the Hodge dual used to compute the regressive product in practice?
5. What is the significance of the degree of the resulting multivector in the regressive product?

### Tip:
The Hodge dual is a powerful tool for relating complementary dimensions in a vector space, especially when working with differential forms or geometric algebra.

Explore the regressive product $ \mathbf{A} \vee \mathbf{B} $ in the Grassmann-Cayley algebra, a fundamental operation in multilinear and geometric algebra. Learn its definition, properties (antisymmetry, associativity, distributivity), and applications. Suitable for advanced mathematics enthusiasts.

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