Contents 1. Introduction 2 2. Preliminaries 4 2.1. Some results on ...

175and the associative conditions holdA µ Q ◦ ( σ A Q ) = µ B Q ◦ ( Qσ B) and B µ P ◦ ( σ B P ) = µ A P ◦ ( P σ A) .In fact, we compute[σ A ◦ (A µ Q P )] (− ⊗ R f ⊗ T x ⊗ R a) = σ A (− ⊗ R f ⊗ T xa)= − ⊗ R f (xa) = − ⊗ R f (x) aand[mA ◦ ( Aσ A)] (− ⊗ R f ⊗ T x ⊗ R a) = m A (− ⊗ R f (x) ⊗ R a) = − ⊗ R f (x) aso thatσ A ◦ (A µ Q P ) = m A ◦ ( Aσ A) .We compute[σ A ◦ ( )]Qµ A P (− ⊗R a ⊗ R f ⊗ T x) = σ A (− ⊗ R af ⊗ T x) = − ⊗ R af (x)and[mA ◦ ( σ A A )] (− ⊗ R a ⊗ R f ⊗ T x) = m A (− ⊗ R a ⊗ R f (x)) = − ⊗ R af (x)so that we getσ A ◦ ( Qµ A P)= mA ◦ ( σ A A ) .We compute[σ B ◦ (B µ P Q )] (− ⊗ T x ⊗ R f ⊗ T b) = σ B (− ⊗ T x ⊗ R fb)= σ B (− ⊗ T x ⊗ R f (b ())) = − ⊗ T x · f (b ())[mB ◦ ( Bσ B)] (− ⊗ T x ⊗ R f ⊗ T b) = m B (− ⊗ T x · f () ⊗ T b) = − ⊗ T [(x · f ()) ◦ b]Let us compute, for every y ∈ Σ we have[− ⊗ T x · f (b ())] (y) = − ⊗ T xf (b (y)) =− ⊗ T [(x · f ()) ◦ b] (y) = − ⊗ T (x · f ()) (b (y)) = − ⊗ T xf (b (y))so that we obtainσ B ◦ (B µ P Q ) = m B ◦ ( Bσ B) .Now we compute[σ B ◦ ( )]P µ B Q (− ⊗T b ⊗ T x ⊗ R f) = σ B (− ⊗ T b (x) ⊗ R f) = − ⊗ T b (x) · f ()and[mB ◦ ( σ B B )] (− ⊗ T b ⊗ T x ⊗ R f) = m B (− ⊗ T b ⊗ T x · f ()) = − ⊗ T [b ◦ (x · f ())]so that, for every y ∈ Σ we haveand[− ⊗ T b (x) · f ()] (y) = − ⊗ T b (x) f (y)(− ⊗ T [b ◦ (x · f ())]) (y) = − ⊗ T b (x · f () (y)) = − ⊗ T b (xf (y)) = − ⊗ T b (x) f (y)so that we getσ B ◦ ( P µ B Q)= mB ◦ ( σ B B ) .