The main theorem in this paper is quite rich in applications (not fully exploited here, but rather in a forthcoming paper by the same authors). In order to state it, we outline some definitions. A graded poset $p$ is defined in the paper, along with a function $f\sb J(p)$ that has as output the maximal chains in $p$ with descent set contained in $J$. The algebra $Q\sb \bullet$ of quasi-symmetric functions consists of all formal power series of bounded degree in commuting indeterminates $x\sb 1,x\sb 2,\cdots $ in which the coefficient of $x\sb {i\sb 1}\sp {\alpha\sb 1}x\sb {i\sb 2}\sp {\alpha\sb 2}\cdots x\sb {i\sb k}\sp {\alpha\sb k}$ depends only on $\alpha$ and not on $i\sb 1,\cdots ,i\sb k$ if $i\sb 1<i\sb 2<\cdots<i\sb k$. $Q\sb \bullet$ has a basis of monomial quasi-symmetric functions $M\sb \alpha$. The main object of study here is the function $F\sb p=\sum\sb {\alpha}f\sb {i(\alpha)}(p)M\sb \alpha$. Let $P$ be a class of graded posets closed under taking subintevals and products. The incidence coalgebra $IP$ of $P$ is the graded free abelian group generated by isomorphism classes of posets in $P$ with grading induced by the rank of a poset and coproduct by $\Delta(p)=\sum\sb {x\in p}[\widehat{0},x]\otimes[x,\widehat{1}].$ An algebra structure on $IP$ is defined.

The main theorem in this paper states that if $P$ is a class of edge-labeled posets closed under subintervals and products, then the map $\Phi\colon IP\rightarrow Q\sb \bullet$ induced by $p\in P\rightarrow F\sb {p}\in Q\sb \bullet$ is a morphism of graded Hopf algebras. A short example using the Boolean poset follows the theorem.

The last two sections apply some of the previous results to rank-selected posets and symmetric edge-labeled posets.