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K − 1}, and for n ˜ j ∈ N, X n ˜j Rn˜ j ×nj , where Z 1 , . . , Z j−1 do not show up for j = 1, and Z j+2 , . . , Z k do not show up for j = k − 1; (3Xk ) ˜ k, X k Tk = Tk X if then ˜ k )(Z 1 , . . , Z k−1 , Z k Tk ) f (X , . . , X )(Z , . . , Z k )Tk = f (X 0 , . . , X k−1 , X 0 k 1 ˜ k ∈ Ω , and Tk ∈ Rnk ×˜nk , where Z 1 , . . , Z k−1 do not show up for n ˜ k ∈ N, X n ˜k when k = 1. Moreover, conditions (1X0 ) and (2X0 ) in the deﬁnition of a nc function of order k together are equivalent to condition (3X0 ), and similarly, (1Xj ) & (2Xj ) ⇐⇒ (3Xj ) (j = 1, .

Proof. Trivial. 4. Let f : Ω → Nnc , g : Ω → Onc be nc functions on a right (respectively, left) admissible nc set Ω ⊆ Mnc . , we are given a R-linear map from N ⊗R O to P). We extend the product operation to matrices over N and over O of appropriate sizes. It is easy to check that f · g : Ω → Pnc is a nc function. Then ΔR (f · g)(X, Y )(Z) = f (X) · ΔR g(X, Y )(Z) + ΔR f (X, Y )(Z) · g(Y ) for all n, m ∈ N, X ∈ Ωn , Y ∈ Ωm , and Z ∈ Mn×m (respectively, ΔL (f · g)(X, Y )(Z) = ΔL f (X, Y )(Z) · g(X) + f (Y ) · ΔL g(X, Y )(Z) for all n, m ∈ N, X ∈ Ωn , Y ∈ Ωm , and Z ∈ Mm×n ).

Z k−1 , row [Z k , Z k ]) = row [f (X 0 , . . , X k−1 , X k )(Z 1 , . . , Z k−1 , Z k , f (X 0 , . . , X k−1 , X k )(Z 1 , . . , Z k−1 , Z k )] (k) for nk , nk ∈ N, X k ∈ Ωn , X k Nk nk−1 ×nk k 1 , where Z , . . , Z k−1 ∈ Ωn , Z k ∈ Nk nk−1 ×nk , Z k (k) ∈ k do not show up when k = 1. (0) (k) • f respects similarities: if n0 , . . , nk ∈ N, X 0 ∈ Ωn0 , . . , X k ∈ Ωnk , Z 1 ∈ N1 n0 ×n1 , . . , Z k ∈ Nk nk−1 ×nk , then (2X0 ) f (S0 X 0 S0−1 , X 1 , . . , X k )(S0 Z 1 , Z 2 , . . , Z k ) = S0 f (X 0 , .