Hopf algebras of set systems

Hopf algebras play a major role in various mathematical fields such as algebraic topology, formal group theory, and theoretical physics, and they are gaining significance in combinatorics influenced by G.-C. Rota and his school. This article aims to build upon W. Schmitt's work by establishing combinatorial models for several Hopf algebras associated with the universal formal group law and the Lazard ring. To achieve this, the authors incorporate and extend certain invariants of simple graphs, including the umbral chromatic polynomial and R. Stanley's symmetric function. The main combinatorial elements are finite set systems, along with a versatile generalization where these systems are equipped with a group of automorphisms. The interactions with the Roman-Rota umbral calculus over graded rings, which may include torsion, are significant aspects of this study. Through the introduction of various Hopf algebras of set systems, the authors produce examples with richer algebraic structures that reflect their combinatorial origins, allowing for a comprehensive understanding of the algebraic invariants related to these systems.