Suppose I have a set of N vectors $p_0$ belong to a certain vector algebra. There is a natural "follow-up" algebra obtained from this set defined by the $N^2$ elements $p_i p_j$ (actually $N(N+1)/2$ elements if the algebra has a commutative multiplication). We can iterate and obtain all these "follow-up" sets until we arrive to a single element defined by $\prod^N_i{p_i}$
What is the standard terminology for these follow-up algebra constructions in the literature?