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In this paper we study of the structure of non-commutative Poisson algebraswith an arbitrary set $\ss.$ We show that any of such an algebra $\pp$decomposes as$$\pp=\uu\oplus\sum_{[\lambda]\in(\Lambda_\ss\setminus\{0\})/\sim}\pp_{[\lambda]},$$where $\uu$ is a linear subspace complement of $\span_{\bbbf}\{ [\pp_{\mu},\pp_{\eta}]+\pp_{\mu}\pp_{\eta} : \mu, \eta\in[\lam]\}\cap\pp_0$ in $\pp_0$ andany $\pp_{[\lambda]}$ a well-described graded ideals of $\pp,$ satisfying$[\pp_{[\lambda]}, \pp_{[\mu]}]+\pp_{[\lambda]} \pp_{[\mu]}=0$ if$[\lambda]\neq[\mu].$  Under certain conditions, the simplicity of $\pp$ is characterized and it isshown that $\pp$ is the direct sum of the family of its graded simple ideals.

Non-commutative Poisson algebras with a set grading