Proposition 10

— Let $f \in A\{I\}$; then the following conditions are equivalent : (i) $f$ is invertible in $A\{I\}$ for the law $\circ$; (ii) $T_f$ is invertible in the ring $\mathrm{End}(A^{(I)})$.

The implication (i) $\Rightarrow$ (ii) is immediate from (28). Suppose now that $T_f$ is invertible in $\mathrm{End}(A^{(I)})$. There exists $g = (g_i)_i \in A\{I\}$ such that each $g_i$ is homogeneous of degree $l$ and $T_g \circ T_f$ is the identity mapping of $A^{(I)}$. Write $h = g \circ f$; then (28) shows that $T_h$ is the identity mapping of $A^{(I)}$, which is equivalent to the assertion $\omega(h_i - X_i) \geq 2$. By Lemma 2 of IV, p. 35 $h$ is therefore invertible in $A\{I\}$. It is clear that $g$ is invertible in $A\{I\}$, hence $f$ is invertible in $A\{I\}$.