Some remarks for the Akivis algebras and the Pre-Lie algebras

In this paper, by using the Composition-Diamond lemma for non-associative algebras invented by A. I. Shirshov in 1962, we give Gr"{o}bner-Shirshov bases for free Pre-Lie algebras and the universal enveloping non-associative algebra of an Akivis algebra, respectively. As applications, we show I.P. Shestakov’s result that any Akivis algebra is linear and D. Segal’s result that the set of all good words in $X^{\ast \ast }$ forms a linear basis of the free Pre-Lie algebra $PLie(X)$ generated by the set $X$. For completeness, we give the details of the proof of Shirshov’s Composition-Diamond lemma for non-associative algebras.