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.
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.
Mother Law \(\sigma^+\circ 0=\sigma \circ \sigma^+ \)
\({\rm Grp}_{\rm pt} ({\rm RK}J,G)\cong \mathbb N{\rm Set}_{\rm pt} (J, \Sigma^G)\)
