  The theory of operated algebras has played a pivotal role in mathematics and
physics. In this paper, we introduce a $\lambda$-TD algebra that appropriately
includes both the Rota-Baxter algebra and the TD-algebra. The explicit
construction of free commutative $\lambda$-TD algebra on a commutative algebra
is obtained by generalized shuffle products, called $\lambda$-TD shuffle
products. We then show that the free commutative $\lambda$-TD algebra possesses
a left counital bialgera structure by means of a suitable 1-cocycle condition.
Furthermore, the classical result that every connected filtered bialgebra is a
Hopf algebra, is extended to the context of left counital bialgebras. Given
this result, we finally prove that the left counital bialgebra on the free
commutative $\lambda$-TD algebra is connected and filtered, and thus is a left
counital Hopf algebra.
