Commutators of Pre-Lie n-Algebras and PL∞-Algebras

We show that a PL∞-algebra V can be described by a nilpotent coderivation of degree −1 on coalgebra P*V. Based on this result, we can generalise the result of Lada to show that every A∞-algebra carries a PL∞-algebra structure and every PL∞-algebra carries an L∞-algebra structure. In particular, we obtain a pre-Lie n-algebra structure on an arbitrary partially associative n-algebra and deduce that pre-Lie n-algebras are n-Lie admissible.