Combinatorial Hopf algebras from restriction species with preorder cuts
Gunnar Fl{\o}ystad

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 whose objects are sets, but morphisms are represented by matrices of natural numbers, and 2. restriction species on sets coming with pairs of natural transformations Pre to the species of preorders. These induce two coproducts and . Dualizing gives product and coproduct , giving bimonoid species.
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