Free pre-lie algebras of finite posets

TL;DR

This paper constructs a free pre-Lie algebra structure on finite posets and explores its duality with a non-associative permutative coalgebra, extending results to finite connected topological spaces.

Contribution

It introduces a novel free pre-Lie algebra framework for finite posets and establishes a duality with a non-associative permutative coalgebra, extending to topological spaces.

Findings

Finite posets form a free pre-Lie algebra.

A non-associative permutative coproduct is constructed.

Duality between the product and coproduct is established.

Abstract

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.