The chromatic symmetric function on graphs is a celebrated graph invariant. Analogous chromatic maps can be defined on other objects, as presented by Aguiar, Bergeron and Sottile. The problem of identifying the kernel of some of these maps was addressed by Féray, for the Gessel quasisymmetric function on posets.

On graphs, we show that the modular relations and isomorphism relations span the kernel of the chromatic symmetric function. This helps us to construct a new invariant on graphs, which may be helpful in the context of the tree conjecture. We also address the kernel problem in the Hopf algebra of generalized permutahedra, introduced by Aguiar and Ardila. We present a solution to the kernel problem on the Hopf algebra spanned by hypergraphic polytopes, which is a subfamily of generalized permutahedra that contains a number of polytope families.

Finally, we consider the non-commutative analogues of these quasisymmetric invariants, and establish that the word quasisymmetric functions, also called non-commutative quasisymmetric functions, form the terminal object in the category of combinatorial Hopf monoids. As a corollary, we show that there is no combinatorial Hopf monoid morphism between the combinatorial Hopf monoid of posets and that of hypergraphic polytopes.

图上的色对称函数是著名的图不变性。可以在其他对象上定义类似的色图，如Aguiar，Bergeron和Sottile所介绍的。费雷（Féray）解决了确定某些图的核的问题，这是针对姿态集上的Gessel拟对称函数的。

在图上，我们表明模数关系和同构关系跨越了色对称函数的核。这有助于我们在图上构造新的不变量，这可能在树状猜想的背景下有所帮助。我们还解决了由Aguiar和Ardila引入的广义置换的Hopf代数中的内核问题。我们提出了关于由超图多面体跨越的Hopf代数上的核问题的解决方案，它是一个包含许多多面体家族的广义排列变体的子族。

最后，我们考虑了这些拟对称不变量的非交换类似物，并确定了拟对称函数（也称为非交换拟对称函数）一词构成了组合Hopf单面体类别的最终对象。作为推论，我们证明了在波普特的组合霍普夫单义素和超图多义词之间没有组合的霍普夫单义素态。