Remark

(1) We shall later define an isomorphism of Libas_A(I) onto the tensor algebra of the free module $A^{(I)}$ (§ 5, no. 5) and also an isomorphism of Libasc_A(I) onto the symmetric algebra of $A^{(I)}$ (§ 6, no. 6).

(2) Let $\rho$ be a unital homomorphism of A into a commutative ring B. As has been seen (§ 2, no. 6), an isomorphism $\sigma$ is derived from $\rho$ of Lib_B(I) (resp. Libas_B(I), resp. Libasc_B(I)) onto the algebra $(\mathrm{Lib}_A(I))_{(B)}$ (resp. $(\mathrm{Libas}_A(I))_{(B)}$, resp. $(\mathrm{Libasc}_A(I))_{(B)}$) obtained by extending the scalars to B by means of $\rho$; if $X_i^A, X_i^B$ are the indeterminates of index $i$ corresponding respectively to A and B, then $\sigma(X_i^B) = X_i^A \otimes 1$.

(3) Let J be a subset of I; we know that M(J) is identified with a stable subset of the magma M(I) and hence (no. 6) Lib_A(J) is canonically identified with a subalgebra of Lib_A(I), generated by the $X_i$ such that $i \in J$; it is said that only the indeterminates of indices belonging to J occur in an element of Lib_A(J). The definition given in no. 6 of the algebra of a magma shows that Lib_A(I) is the union of the directed family of subalgebras Lib_A(J) when J runs through the set of finite subsets of I. There are analogous results for Libas_A(I) and Libasc_A(I).

(4) With element s of M(I) (resp. Mo(I), resp. $\mathbf{N}^{(I)}$) is associated its length $l(s)$, which is an integer $\geqslant 1$ (resp. $\geqslant 0$) such that $l(ss') = l(s) + l(s')$ (I, § 7, nos. 1, 2 and 7). If $e_s$ is the element of Lib_A(I) (resp. Libas_A(I), resp. Libasc_A(I)) corresponding to s, the total degree (or simply the degree of an element $x = \sum_s \alpha_s e_s \neq 0$ of Lib_A(I) (resp. Libas_A(A), resp. Libasc_A(I)) is the greatest of the numbers $l(s)$ when s runs through the (non-empty by hypothesis) set of elements such that $\alpha_s \neq 0$. For example, if $i, j, k$ are three distinct elements of I, the element $(X_i(X_jX_k))X_i - (X_iX_j)(X_kX_i)$ is an element $\neq 0$ of total degree 4 in Lib_A(I).