LSM is not generated by binary functions

We will construct a set C containing binary lsm functions and closed under a minimalist set of operations related to T2-constructibility [3]. Later, in section 3, we will relate this to ppsω-definability. For concision we will use vector notation such as x = (x1, · · · , xn), (x,y) = (x1, · · · , xn, y1, · · · , ym), 1 = (1, · · · , 1), (i, j,x) = (i, j, x1, · · · , xn), and x · y = x1y1 + · · · + xnyn. For all k, l ≥ 0 and all F ∈ Bk and G ∈ Bl, define the tensor product F ⊗G ∈ Bk+l by (F ⊗G)(x1, · · · , xk, y1, · · · , yl) = F (x1, · · · , xk)G(y1, · · · , yl) For all k ≥ 2 and all F ∈ Bk define the primitive contraction tr1,2F ∈ Bk−2 by