Leibniz-Hopf Algebra And Quasi-Symmetric Functions

Let $\mathcal{M}$ be the graded dual of the Leibniz–Hopf algebra over the integers. The strong Ditters conjecture states that $\mathcal{M}$ is a free commutative algebra with as generators the concatenation powers of elementary Lyndon words. This conjecture is still open (as of 2001); the initial proof contains mistakes (so the assertion of its proof in Leibniz–Hopf algebra is incorrect), and so does a later version [a1] of it. Meanwhile, the weak Ditters conjecture, which states that $\mathcal{M}$ is free over the integers without giving a concrete set of generators, has been proved; see Quasi-symmetric function and [a2].

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