Homotopy Gerstenhaber structure on deformation complex of a morphism

$G_{\mathrm{\infty }}$-structure is shown to exist on the deformation complex of a morphism of associative algebras. The main step of the construction is extension of a $B_{\mathrm{\infty }}$-algebra by an associative algebra. Actions of $B_{\mathrm{\infty }}$-algebras on associative and $B_{\mathrm{\infty }}$-algebras are analyzed, extensions of $B_{\mathrm{\infty }}$-algebras by associative and $B_{\mathrm{\infty }}$-algebras, that they act upon, are constructed. The resulting $G_{\mathrm{\infty }}$-algebra on the deformation complex of a morphism is shown to be quasi-isomorphic to the $G_{\mathrm{\infty }}$-algebra on deformation complex of the corresponding diagram algebra.