Proposition 11

("associativity" of the tensor $\varepsilon$-product). In the notation of Proposition 10, let $(J_\lambda)_{\lambda \in L}$ be a partition of I and write $\Delta'_\lambda = \prod_{i \in J_\lambda} \Delta_i$ for all $\lambda \in L$. Let $E'_\lambda$ be a graded tensor $\varepsilon$-product of type $\Delta'_\lambda$ of the family $(E_i)_{i \in J_\lambda}$ for some total ordering chosen on $J_\lambda$. On the other hand, for $\lambda, \mu$ in $L$ and $\lambda \neq \mu$, we write, for $\alpha'_\lambda = (\alpha_i)_{i \in J_\lambda}$, $\beta'_\mu = (\beta_j)_{j \in J_\mu}$,

Then $(\varepsilon'_{\lambda \mu})$ is a system of commutation factors over the $\Delta'_\lambda$ with values in $A$ and there exists one and only one homomorphism of graded algebras of type $\Delta$, $v : \varepsilon' \bigotimes_{\lambda \in L} E'_\lambda \to \varepsilon \bigotimes_{i \in I} E_i$, such that

for all $(x_i) \in \prod_{i \in I} E_i$, provided that the total ordering is taken on I which induces on each $J_\lambda$ the chosen total ordering, and which is such that, for $\lambda < \mu$ in $L$, $i \in J_\lambda$ and $j \in J_\mu$, $i < j$.

The fact that the $\varepsilon'_{\lambda \mu}$ form a system of commutation factors is trivial. Let $h_{i,\lambda} : E_i \to E'_\lambda$, $h'_\lambda : E'_\lambda \to \varepsilon' \bigotimes_{\lambda \in L} E'_\lambda$ be the canonical homomorphisms (for $\lambda \in L$, $i \in J_\lambda$) and write $h''_i = h'_\lambda \circ h_{i,\lambda}$; it will suffice by virtue of the uniqueness of the solution of a universal mapping problem, to show that $\varepsilon' \bigotimes_{\lambda \in L} E'_\lambda$ and the $h''_i$ satisfy the conditions of Proposition 10. Now, for all $\lambda \in L$, let $f'_\lambda : E'_\lambda \to F$ be the unique algebra homomorphism such that $f'_\lambda \circ h_{i,\lambda} = f_i$ for all $i \in J_\lambda$. We show that, for $\lambda \neq \mu$, $\alpha'_\lambda = (\alpha_i)_{i \in J_\lambda}, \beta'_\mu = (\beta_j)_{j \in J_\mu}$,

for $x'_\lambda \in E'_\lambda$ (resp. $x'_\mu \in E'_\mu$) homogeneous of degree $\alpha'_\lambda$ (resp. $\beta'_\mu$); it suffices, by linearity, to verify it when $x'_\mu = \bigotimes_{i \in J_\lambda} x_i, x'_\mu = \bigotimes_{j \in J_\mu} x_j, x_i$ (resp. $x_j$) being homogeneous of degree $\alpha_i$ (resp. $\beta_j$) in $E_i$ (resp. $E_j$) for $i \in J_\lambda, j \in J_\mu$. But this follows from formula (30) which defines the $f'_\lambda$ and Lemma 3 of no. 6, taking account of hypothesis (26) and definition (32). There is therefore one and only one algebra homomorphism $f : \varepsilon' \bigotimes_{\lambda \in L} E'_\lambda \to F$ such that $f \circ h'_\lambda = f'_\lambda$ for all $\lambda \in L$; whence $f \circ h''_i = f_i$ for all $i \in I$ and the uniqueness of $f$ is trivial.