Poisson structure on manifolds with corners

Since a Poisson Structure is a smooth bivector field, we use a ring-valued sheaf ${\text{OO}}_X$ on a manifold with corners $X$, we can interpret ${\text{OO}}_X(U)$ as the ring of admissible smooth functions where $U$ is an open subset on $X$, in this way, a poisson structure on $(X,{\text{OO}}_X)$ is a sheaf morphism $$\{-,-\}:{\text{OO}}_X\times {\text{OO}}_X⟶{\text{OO}}_X$$ which satisfies the Leibniz rule an also the Jacobi Identity.