  In this paper, we first recall the construction of a twisted pre-Lie algebra
structure on the species of finite connected topological spaces. Then we
construct the corresponding nonassociative permutative coproduct, and we prove
that the vector space generated by isomorphism classes of finite posets is a
free pre-Lie algebra and is a co-free non-associative permutative coalgebra. In
the end, we give an explicit duality between the non-associative permutative
product and the proposed non-associative permutative coproduct. Finally, we
prove that the results in this paper remain true for the finite connected
topological spaces.
