A Hopf Monoid On Set Families

Abstract

We first introduce a Hopf monoid on set families called SF. Following that we will use the topological methods of Aguiar and Ardila to find a cancellation-free formula for the Hopf submonoid of SF spanned by lattices of order ideals. We will then turn our attention to the Hopf submonoid spanned by simplicial complexes in which we derive an antipode formula for simplex skeletons.We then turn our attention to the Hopf submonoid of SF spanned by chain gangs. The character group of this submonoid is related to formal power series. We proceed to show that the Hopf algebra of symmetric functions is a quotient of the Hopf algebra of chain gangs. Finally we conclude with suggestions for future research directions in the study of SF.