Corollary 1

— Let $\Gamma$ be a monoid, L a field and X a set of homomorphisms of $\Gamma$ into the multiplicative monoid of L. Then X is a free subset of the vector L-space $L^\Gamma$ of mappings of $\Gamma$ into L.

Let A be the algebra of the monoid $\Gamma$ with coefficients in L and $(e_\gamma)_{\gamma \in \Gamma}$ the canonical basis of A over L (III, p. 446). For every L-linear mapping $u$ of A into L let us write $\tilde{u}(\gamma) = u(e_\gamma)$ (for $\gamma \in \Gamma$); then the mapping $u \mapsto 3$ is an isomorphism of vector L-spaces of $\mathrm{Hom}_L(A, L)$ onto $L^\Gamma$ which maps onto X the set of L-algebra homomorphisms of A into L. Now it suffices to apply Th. 1 with $K = L$.