Remark

(1) We recover the tensor product of algebras defined in no. 1 (with moreover the graduation the tensor product of those of its factors) taking $\varepsilon_{ij}(\alpha_i, \beta_j) = 1$ for all $i, j, \alpha_i$ and $\beta_j$.

(2) Suppose that all the $\Delta_i$ are equal to $\mathbf{Z}$ and write $\varepsilon_{ij}(\alpha_i, \beta_j) = (-1)^{\alpha_i \beta_j}$; the tensor $\varepsilon$-product $^{\varepsilon} \bigotimes_{i \in I} E_i$ corresponding to this system of commutation factors is then called the skew tensor product of the graded algebras $E_i$ of type $\mathbf{Z}$ and denoted by $^{\varepsilon} \bigotimes_{i \in I} E_i$ (or $E \ ^{\varepsilon} \otimes_A F$ for two algebras, or $^{\varepsilon} G^{\otimes n}$ instead of $^{\varepsilon} G^{\otimes n}$).