摘要： We introduce the Hopf algebra of quasi-symmetric functions with semigroup exponents generalizing the Hopf algebra QSym of quasi-symmetric functions. As a special case we obtain the Hopf algebra WQSym of weak quasi-symmetric functions, which provides a framework for the study of a question proposed by G.-C. Rota relating symmetric type functions and Rota?Baxter algebras. We show that QSym is a Hopf subalgebra and a Hopf quotient algebra of WQSym. Rota’s question is addressed by identifying WQSym with the free commutative unitary Rota?Baxter algebra of weight 1 on one generator, which also allows us to equip the latter with a Hopf algebra structure. This is joint work with Li Guo, Jean-Yves Thibon, and Jianqiang Zhao.