View source for Leibniz-Hopf algebra and quasi-symmetric functions

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130030/l1300301.png" /> be the graded dual of the [[Leibniz–Hopf algebra|Leibniz–Hopf algebra]] over the integers. The strong Ditters conjecture states that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130030/l1300302.png" /> 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|Leibniz–Hopf algebra]] is incorrect), and so does a later version [[#References|[a1]]] of it. Meanwhile, the weak Ditters conjecture, which states that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130030/l1300303.png" /> is free over the integers without giving a concrete set of generators, has been proved; see [[Quasi-symmetric function|Quasi-symmetric function]] and [[#References|[a2]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.J. Ditters,   A.C.J. Scholtens,   "Free polynomial generators for the Hopf algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130030/l1300304.png" /> of quasi-symmetric functions"  ''J. Pure Appl. Algebra'' , '''144'''  (1999)  pp. 213–227</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Hazewinkel,   "Quasi-symmetric functions"  D. Krob (ed.)  A.A. Mikhalev (ed.)  A.V. Mikhalev (ed.) , ''Formal Power Series and Algebraic Combinatorics (Moscow 2000)'' , Springer  (2000)  pp. 30–44</TD></TR></table>