AI 中文总结
研究自由幺半群上代表性函数的因式分解与分解，借助Kleene - Schützenberger定理将其转化为图的分解，通过研究非交换级数的乘积和余积来有效分解图，得出相关双代数同构于特定多项式双代数的Sweedler对偶的结论。
AI 中文摘要
自由幺半群\(X^*\)（由字母表\(X\)生成）且取值于包含有理数域\(\mathbb{Q}\)的环\(A\)中的代表性函数的因式分解和分解，等同于其图（在\(X\)上的有理非交换级数的\(A -\)代数内）的因式分解和分解（借助Kleene - Schützenberger定理）。为有效分解这些图，我们研究了各种非交换级数的乘积（如连接、洗牌及其\(\phi\)变形）和余积。当\(A\)为域\(K\)时，其相关的非分次交换和余非交换级数双代数同构于具有仅平面的Kleene星作为特征的分次非交换余交换多项式双代数的Sweedler对偶，或者等价地，仅平面是无穷小特征（借助类似Ree定理）。
英文摘要
Factorization and decomposition of representative functions on a free monoid X * (generated by an alphabet X ) and with values in a ring A containing Q are equivalent to factorization and decomposition of their graphs (within the A-algebra of rational noncommutative series over X ) admitting linear representations (thanks to the Kleene-Sch{ü}tzenberger theorem). To factorize and to decompose effectively these graphs, we examine various products of noncommutative series (as concatenation, shuffle and its $ϕ$deformations) and co-products such that, for A is a field K, their associated non graded commutative and co-noncommutative bialgebras of series are isomorphic to the Sweedler's dual of the graded noncommutative co-commutative bialgebras of polynomials having, for the concatenation, only Kleene stars of the planes as characters, or equivalently, only the planes are infinitesimal characters (thanks to a Ree's theorem like).
