Realizations of Hopf algebras of graphs by alphabets

Abstract: We here give polynomial realizations of various Hopf algebras or bialgebras on Feynman graphs, graphs, posets or quasi-posets, that it to say injections of these objects into polynomial algebras generated by an alphabet. The alphabet here considered are totally quasi-ordered. The coproducts are given by doubling the alphabets; a second coproduct is defined by squaring the alphabets, and we obtain cointeracting bialgebras in the commutative case.