Combinatorial Hopf algebras from restriction species with preorder cuts

TL;DR

This paper introduces new Hopf algebras derived from restriction species and preorder cuts, expanding the algebraic structures related to permutations, parking filtrations, and preorders.

Contribution

It develops a categorical framework with restriction species and dual coproducts, leading to novel Hopf algebras and bimonoid species.

Findings

Constructed quotient Hopf algebras of the Malvenuto-Reutenauer algebra.

Defined a category Set_N with matrix morphisms for sets.

Introduced restriction species with pairs of natural transformations to preorders.

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.