Free pre-Lie algebras of finite posets

We first recall the construction of a twisted pre-Lie algebra structure on the species of finite connected topological spaces. Then, we construct a corresponding non-coassociative permutative (NAP) coproduct on the subspecies of finite connected T0 topological spaces, i.e. finite connected posets, and we prove that the vector space generated by isomorphism classes of finite posets is a free pre-Lie algebra and also a cofree NAP coalgebra. Furthermore, we give an explicit duality between the non-associative permutative product and the proposed NAP coproduct. Finally, we prove that the results in this paper remain true for finite connected topological spaces.