Non-commutative Poisson algebras with a set grading

In this paper we study of the structure of non-commutative Poisson algebras with an arbitrary set $\text{ß}\mathrm{.}$ We show that any of such an algebra $\text{pp}$ decomposes as $$\text{pp}=\text{uu}\oplus \sum _{[\lambda ]\in ({\mathrm{\Lambda }}_{\text{ß}}\setminus \{0\})\mathrm{/}\sim }{\text{pp}}_{[\lambda ]},$$ where $\text{uu}$ is a linear subspace complement of ${\text{span}}_{\text{bbbf}}\{[{\text{pp}}_{\mu },{\text{pp}}_{\eta }]+{\text{pp}}_{\mu }{\text{pp}}_{\eta }:\mu ,\eta \in [\text{lam}]\}\cap {\text{pp}}_0$ in ${\text{pp}}_0$ and any ${\text{pp}}_{[\lambda ]}$ a well-described graded ideals of $\text{pp},$ satisfying $[{\text{pp}}_{[\lambda ]},{\text{pp}}_{[\mu ]}]+{\text{pp}}_{[\lambda ]}{\text{pp}}_{[\mu ]}=0$ if $[\lambda ]\mathrm{\ne }[\mu ]\mathrm{.}$ Under certain conditions, the simplicity of $\text{pp}$ is characterized and it is shown that $\text{pp}$ is the direct sum of the family of its graded simple ideals.