Some scope for things such as \[2p + 2p = 4p^2 \\ \\ \begin{bmatrix} 1 \\ 0 \end{bmatrix} + \begin{bmatrix} 1 & 0 \end{bmatrix} \to \begin{bmatrix} \begin{bmatrix} 1 & 0 \end{bmatrix} \\ 0 \end{bmatrix} \to \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}\]
But this system appears to break down in many circumstances. For this particular example, one could continue to add \(2p\) and generate \(2^np^n\) like structures of rank \(n\) quite comfortably.
Obviously this is the overlap of product and multiplication, as this is still the dyadic product. something else is needed, some other operation.