Various products of representative series

For factorizing and decomposing the representative (or rational) series, with coefficients in a commutative ring <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>A</mml:mi> </mml:math> containing <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ℚ</mml:mi> </mml:math> , we examine various products (such as concatenation, shuffle, quasi-shuffle) defined on the free monoid which are such that their associated bialgebras are isomorphic to the Sweedler’s dual, for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>A</mml:mi> </mml:math> being a field <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>K</mml:mi> </mml:math> .