Looking for constructive feedback on this. I don't know much abstract algebra so please take that into account when you give feedback.
Suppose one wants to map two class structures sitting in $\Bbb R^3,$ from $\Bbb R^3$ to $\Bbb R^2,$ via projection.
Here $f$ represents a transformation (in this case a projection) that acts on the class structures $\Phi$ and $M.$ $S$ acts to generate the class structures respectively.
$f:\operatorname{class}\{\Phi_S(X)\} \to \operatorname{class}\{\Phi_S(X)\},$ via $f:= \Bbb P_\Phi:\Bbb R_\Phi^3\to \Bbb R_\Phi^2.$
$f^*:\operatorname{class}\{M_S(X)\}\to \operatorname{class}\{M_S(X)\},$ via $f^*:= \Bbb P_M: \Bbb R_M^3\to \Bbb R_M^2. $
That notation just describes the class structures being projected from $\Bbb R^3\to \Bbb R^2.$
Intersecting either class structure with a plane yields a ring structure. I'm interested in describing such a ring structure after the projection. Is using a graph a useful way? The reason I'm thinking that, is because the following graph encodes information about the ring structures, via intersections in a lower dimension than three.
Note: The graph includes four class structures mapped down to $\Bbb R^2,$ not just two. I denote the class structures as $\operatorname{class}\{\Phi_S(X)\}=(E,G)$ and $\operatorname{class}\{M_S(X)\}=(H,F).$ Here, the cardinality, or order of $S$ is: $|S|=11,$ because each class structure has $11$ outgoing/incoming edges.
The following is a specific representation of the four class structures sitting in $\Bbb R^3,$ before being mapped down to $\Bbb R^2.$
