Tetration Forum

I conjecture that tetration introduces numbers with fractional dimension \( \mathbb{R}^{\mathbb{R}} = ^2\mathbb{R} \).
For that purpose, I'm trying to extend Geometric algebra (Clifford algebra \( \mathcal{C}\ell_{n,0,0} \)), \( n \in \mathbb{N} \) to fractional dimension, defining \( r \)-blades (of grade \( r \in \mathbb{R} \)).
I attached a PDF explaining it. (it is a draft, not a finished text. And I'm not a mathematician).
In Clifford algebra, given two normal vectors (1-blades), their product should be anticommutative:
\[ f \cdot g = -g \cdot f \]
Each vector can be decomposed into \( s \) fracvectors with fractional dimension \( D = \frac{1}{s} \) (\( s \in \mathbb{N} \)).
\[ f = f_0 \cdot f_1 \cdot f_2 \cdot \ldots \cdot f_s \]
\[ g = g_0 \cdot g_1 \cdot g_2 \cdot \ldots \cdot g_k \]
The problem appears when trying to define the dot product. I don't know how to define the anticommutativity of \( f_i \) with \( g_j \).
\[ f \cdot g = -g \cdot f \]
but
\[ f \cdot g = f \cdot g_0 \cdot g_1 \cdot g_2 \cdot \ldots \cdot g_k \]
then \( f \cdot g = -g \cdot f \) should be preserved regardless of how many \( g_i \) there are (independently of the value of \( s \), \( k \), which determine the number of commutations of \( g_i \) with \( f_j \)).
Commuting
\[ f \cdot (g_0 \cdot g_1) = -g \cdot f \]
Should produce the same signs change than
\[ f \cdot (g_0 \cdot g_1 \cdot g_2) = -g \cdot f \]
regardless of if \( g \) is decomposed in 2 or 3 fractional dimensions
I can't figure how to achieve that. Any idea?

I think that my error was to treat fracvectors as vectors, but they should be treated as Basis n-blades. Basis n-blades are anticommutative only when n is odd.

I don't know how the concept of evenness generalizes to fractional numbers.

Just an algebraic observation. I'm not sure if this can have any geometrical meaning but... have you considered the study of a graded algebraic structures (anticommutative ones graded ring for examples).

In all generality:
Let be a \(G\) group written with additive notation. Assume I have a set of things \(A\) that I want to perform multiplication on. In other words, I assume \(A\) is a monoid, using a multiplicative notation for it.
I also rquire that \(A\) is made by components of various degrees \(\lambda\in G\). This can be said by the two following equivalent conditions:

1) Exists a map \({\rm deg}:A\to G\) assigning to each \(f\in A\) an element of \(G\), its degree;
2) The whole set can be subdivided in subsets of homogeneous degree, \(A=\bigcup_{\lambda \in G}A_d\) where \(A_d\) is the set of elements with the same degree equals \(d\).

On top of this, we want that the operation respects the grading: \[ {\rm deg} (f\cdot g)={\rm deg}(f)+{\rm deg}(g)\] This amounts to asking that multiplying elements \(f\in A_d\) with elements \(f\in A_e\) sends us straight to \(d+e\)-graded elements \(f\cdot g\in A_{d+e}\).

At the end, if I had to define an anticommutation property, I'd like to have a way to generalize the concept of oddness to arbitrary degrees. 
This is usually done via the concept of "signed-group". We have to assign a "sign-rule" to \(G\) where "sign" can be understood as even/odd duality. This means asking an assignement \(\sigma:G\to \mathbb Z/2\mathbb Z\) such that \(\sigma(d+e)=\sigma(d)+\sigma(e) \) where:
 if \(\sigma (d)=0\) we will call thad degree "even", while \(d\) is "odd" if \(\sigma (d)=1\). As you can see, all the rules of the odd/even opposition are respected.

Once we have generalized "oddness" we have to let \( \mathbb Z/2\mathbb Z\) act on \(A\). Roughly as \[f\cdot g= (-1)^{\sigma{\rm deg}(f)\cdot \sigma{\rm deg}(g)}(g\cdot f)\] This can not make sense in full generality because we don't always know what means to evaluate \((-1)f\in A\). Maybe in your particular case we could assume that \(-1\in A_{0_G}\) has the degree corresponding to the zero element (identity) of \(G\).

I believe this can be made more general.

Here's an idea. Let \(A_{x\in \mathbb R} \) a \(\mathbb R\)-graded monoid. \(d:A\to \mathbb R\) satisfies \(d(f\cdot g)=d(f)+d(g)\).

Let \(\mathbb T\) be circle group, whose elements are to be seen as complex numbers \(e^{i\theta}\). Assume \(\mathbb T\) can be represented in \(A_0\) then we are enriching the graded monoid \(A_x\) with an action \(\mathbb T \times A_x\to A_x\) for each \(x\in\mathbb R\).

I wonder if the condition \(\displaystyle g\cdot f= e^{ d(f)d(g) \pi i}\cdot f\cdot g  \) can give us what we want...