Twisted pre-Lie algebras of finite topological spaces

In this paper, we first study the species of finite topological spaces recently considered by Fauvet, Foissy, and Manchon. Then, we construct a twisted pre-Lie structure on the species of connected finite topological spaces. The underlying pre-Lie structure defines a coproduct on the species of finite topological spaces different from those already defined by the authors above. In the end, we illustrate the link between the Grossman-Larson product and the proposed coproduct.