Hopf monoids of set families

TL;DR

This paper introduces a Hopf monoid structure on grounded set families, connecting combinatorial objects like simplicial complexes and matroids, and provides explicit formulas and extensions related to lattices of order ideals and symmetric functions.

Contribution

It defines a new Hopf monoid on grounded set families, extends known algebraic structures, and derives explicit antipode formulas using topological methods.

Findings

A cancellation-free antipode formula for lattices of order ideals.

Extension of the Hopf algebra of lattices of order ideals to symmetric functions.

The character group extends formal power series with constant term 1.

Abstract

A \textit{grounded set family} on $I$ is a subset $F\subseteq2^I$ such that $\emptyset\in F$. We study a linearized Hopf monoid \textbf{SF} on grounded set families, with restriction and contraction inspired by the corresponding operations for antimatroids. Many known combinatorial species, including simplicial complexes and matroids, form Hopf submonoids of \textbf{SF}, although not always with the "standard" Hopf structure (for example, our contraction operation is not the usual contraction of matroids). We use the topological methods of Aguiar and Ardila to obtain a cancellation-free antipode formula for the Hopf submonoid of lattices of order ideals of finite posets. Furthermore, we prove that the Hopf algebra of lattices of order ideals of chain gangs extends the Hopf algebra of symmetric functions, and that its character group extends the group of formal power series in one variable with constant term 1 under multiplication.