Hopf Algebra Structure of Generalized Quasi-Symmetric Functions in Partially Commutative Variables

We introduce a coloured generalization $\mathrm{NSym}_A$ of the Hopf algebra non-commutative symmetric functions described as subalgebra rooted ordered trees algebra. Its natural basis can be identified with set sentences over alphabet $A$ (the colours). present also its graded dual $\mathrm{QSym}_A$ quasi-symmetric together realization in terms power series partially commutative variables. provide formulas expressing multiplication, comultiplication and antipode for these algebras various bases — corresponding generalizations complete homogeneous, elementary, ribbon Schur sum $\mathrm{NSym}$, monomial fundamental $\mathrm{QSym}$. study certain distinguished setting restricted duals to algebras.