Relation among $n$-ary algebras and homotopy algebras

We present $PL_{infty}$-algebras in the form of composition of maps and show that a $PL_{infty}$-algebra $V$ can be described by a nilpotent coderivation on coalgebra $P^*V$ of degree $-1$. Using coalgebra maps among $T^*V$, $wedge^*V$, $P^*V$, we show that every $A_{infty}$-algebra carries a $PL_{infty}$-algebra structure and every $PL_{infty}$-algebra carries an $L_{infty}$-algebra structure. In particular, we obtain a pre Lie $n$-algebra structure on an arbitrary partially associative $n$-algebra and deduce pre Lie $n$-algebras are $n$-Lie admissible.