Combinatorial Deformations of Algebras: Twisting and Perturbations

The framework used to prove the multiplicative law deformation of the algebra of Feynman-Bender diagrams is a $\mathit{\text{twisted}}\mathit{\text{shifted}}\mathit{\text{dual}}\mathit{\text{law}}$ (in fact, twice). We give here a clear interpretation of its two parameters. The crossing parameter is a deformation of the tensor structure whereas the superposition parameters is a perturbation of the shuffle coproduct of Hoffman type which, in turn, can be interpreted as the diagonal restriction of a superproduct. Here, we systematically detail these constructions.