Noncommutative symmetric function

The underlying algebra of the Hopf algebra of noncommutative symmetric functions is the free ring Z⟨Z₁, Z₂,...⟩ generated by non-commuting variables Z₁, Z₂, ...

The coproduct takes Z_n to Σ Z_i ⊗ Z_n–i, where Z₀ = 1 is the identity.

The counit takes Z_i to 0 for i > 0 and takes Z₀ = 1 to 1.

Michiel Hazewinkel showed[2] that a Hasse–Schmidt derivation

${\displaystyle D:A\to A[[t]]}$

on a ring A is equivalent to an action of NSymm on A: the part ${\displaystyle D_{i}:A\to A}$ of D which picks the coefficient of ${\displaystyle t^{i}}$, is the action of the indeterminate Z_i.