The Hopf algebra of posets

Work of Aguiar, Bergeron, and Sottile established the category of "combinatorial Hopf algebras" and showed that the quasisymmetric functions are the terminal object in this category. Stembridge's peak functions form a special Hopf subalgebra of the quasisymmetric functions called the "odd" subalgebra; moreover, the algebra of peak functions is terminal in the category of odd combinatorial Hopf algebras.
We will survey this theory and show how these two important algebras can be easily understood in terms of another combinatorial Hopf algebra: the algebra of (equivalences classes of finite, labeled) posets. This talk is based on work joint with Sam Hsiao.