  The notion of pre-Poisson algebras was introduced by Aguiar in his study of
zinbiel algebras and pre-Lie algebras. In this paper, we first introduce
NS-Poisson algebras as a generalization of both Poisson algebras and
pre-Poisson algebras. An NS-Poisson algebra has an associated sub-adjacent
Poisson algebra. We show that a Nijenhuis operator and a twisted Rota-Baxter
operator on a Poisson algebra deforms the structure into an NS-Poisson algebra.
The semi-classical limit of an NS-algebra deformation and a suitable filtration
of an NS-algebra produce NS-Poisson algebras. On the other hand, F-manifold
algebras were introduced by Dotsenko as the underlying algebraic structure of
F-manifolds. We also introduce NS-F-manifold algebras as a simultaneous
generalization of NS-Poisson algebras, F-manifold algebras and pre-F-manifold
algebras. In the end, we show that Nijenhuis deformations of F-manifold
algebras and the semi-classical limits of NS-pre-Lie algebra deformations have
NS-F-manifold algebra structures.
