Dendriform structures for restriction-deletion and restriction-contraction matroid Hopf algebras

TL;DR

This paper introduces a new Hopf algebra structure on matroids using restriction and deletion, explores dendriform coalgebra structures, and defines a monomial invariant satisfying a convolution identity.

Contribution

It presents a novel Hopf algebra framework for matroids based on restriction and deletion, and initiates the study of dendriform coalgebra structures and invariants.

Findings

New Hopf algebra structure on matroids

Dendriform coalgebra structures on matroids

A monomial invariant satisfying a convolution identity

Abstract

We endow the set of isomorphic classes of matroids with a new Hopf algebra structure, in which the coproduct is implemented via the combinatorial operations of restriction and deletion. We also initiate the investigation of dendriform coalgebra structures on matroids and introduce a monomial invariant which satisfy a convolution identity with respect to restriction and deletion.