Poisson structure on manifolds with corners

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Since a Poisson Structure is a smooth bivector field, we use a ring-valued sheaf ₓ on a manifold with corners X, we can interpret 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, ₓ) is a sheaf morphism {-, -}: ₓ × ₓ ₓ which satisfies the Leibniz rule an also the Jacobi Identity.