Poincare-Birkhoff-Witt theorem for pre-Lie and postLie algebras

We construct the universal enveloping preassociative and postassociative algebra for a pre-Lie and a postLie algebra respectively. We show that the pairs $(\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{L}\mathrm{i}\mathrm{e},\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{A}\mathrm{s})$ and $(\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{t}\mathrm{L}\mathrm{i}\mathrm{e},\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{t}\mathrm{A}\mathrm{s})$ are Poincare-Birkhoff-Witt-pairs, for the first one it’s a reproof of the result of V. Dotsenko and P. Tamaroff.