Twisted pre-lie algebras of finite topological spaces

TL;DR

This paper explores the algebraic structures of finite topological spaces, introducing a new twisted pre-Lie structure and linking it to existing algebraic operations, thereby expanding the theoretical framework of topological space species.

Contribution

It constructs a novel twisted pre-Lie algebra on connected finite topological spaces and defines a new coproduct different from previous models.

Findings

New twisted pre-Lie structure on finite topological spaces

A distinct coproduct linked to the Grossman-Larson product

Enhanced algebraic understanding of finite topological space species

Abstract

In this paper, we first study the species of finite topological spaces recently considered by F. Fauvet, L. Foissy, and D. 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.