Combinatorial Hopf algebras from restriction species with preorder cuts

Abstract

We get new Hopf algebras (HA): 1. A wealth of quotient HA's of the Malvenuto-Reutenauer HA (the Loday-Ronco HA being a special case). They consist of the permutations avoiding an {\it arbitrary} set of permutations without global descents, 2. A HA of pairs of parking filtrations, and 3. Four HA of pairs of preorders. New concepts in this setting are: 1. a category Set$_{{\mathbb N}}$ whose objects are sets, but morphisms are represented by matrices of natural numbers, and 2. restriction species ${\mathsf S}$ on sets coming with pairs of natural transformations $\pi_1, \pi_2 : {\mathsf S} \rightarrow$ Pre to the species of preorders. These induce two coproducts $\Delta_1$ and $\Delta_2$. Dualizing $\Delta_1$ gives product $\mu_1$ and coproduct $\Delta_2$, giving bimonoid species.