On Hopf algebra structures over operads

binary trees T with n leaves K · T This operad is called the operad Cmg of commutative magma algebras. It is not a regular operad. Similarly, for the free operad Γ generated by A(n) = K{αn} (n ≥ 2) with trivial action of Σn, we have Γ(M)(n) = ⊕ abstract reduced trees T with n leaves K · Treduced trees T with n leaves K · T The operad Cmgω := Γ might be called the operad of commutative tree algebras. Example 2.4.9. The non-Σ operad As is a quotient of Mag with respect to the associativity relation ◦ • ◦ •     ◦ ◦ ◦