Combinatorial Hopf algebras from restriction species with preorder cuts

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 ${\pi }_1,{\pi }_2:\mathsf{S}\to$ Pre to the species of preorders. These induce two coproducts ${\mathrm{\Delta }}_1$ and ${\mathrm{\Delta }}_2$. Dualizing ${\mathrm{\Delta }}_1$ gives product ${\mu }_1$ and coproduct ${\mathrm{\Delta }}_2$, giving bimonoid species.