Associators in mould theory

By developing various techniques of mould theory, we introduce $\mathsf{G}\mathsf{A}\mathsf{R}\mathsf{I}(\mathcal{F}{)}_{\mathsf{a}\mathsf{s}+\mathsf{b}\mathsf{a}\mathsf{l}}$, a mould theoretic formulation of Drinfeld’s associator set. We give a mould-theoretical generalization of the result that associator relations imply double shuffle relations, namely, we explain that $\mathsf{G}\mathsf{A}\mathsf{R}\mathsf{I}(\mathcal{F}{)}_{\mathsf{a}\mathsf{s}+\mathsf{b}\mathsf{a}\mathsf{l}}$ is embedded to Ecalle’s set $\mathsf{G}\mathsf{A}\mathsf{R}\mathsf{I}(\mathcal{F}{)}_{\mathsf{a}\mathsf{s}\ast \mathsf{i}\mathsf{s}}$ which is a mould theoretic version of Racinet’s double shuffle set.