Tensor Products and Preservation of Weighted Limits,for S-Posets

We prove that the functor of tensor multiplication by a right S-poset (S is a pomonoid) preserves all small weighted limits if and only if this S-poset is cyclic and projective. We also show that this functor preserves all finite pie-limits if and only if the S-poset is a filtered colimit of S-posets isomorphic to S _S .