In differential geometry a Poisson supermanifold is a differential supermanifold M such that the supercommutative algebra of smooth functions over it (to clarify this: M is not a point set space and so, doesn't "really" exist, and really, this algebra is all we have), is equipped with a bilinear map called the Poisson superbracket turning it into a Poisson superalgebra. Every symplectic supermanifold is a Poisson supermanifold but not vice versa. See also Poisson manifold Poisson algebra Noncommutative geometry Symplectic geometry