Separable Symmetric Tensors and Separable Anti-symmetric Tensors

In this paper, we first initialize the S-product of tensors to unify the outer product, contractive product, and the inner product of tensors. Then, we introduce the separable symmetry tensors and separable anti-symmetry tensors, which are defined, respectively, as the sum and the algebraic sum of rank-one tensors generated by the tensor product of some vectors. We offer a class of tensors to achieve the upper bound for rank(A) <= 6 for all tensors of size 3x3x3. We also show that each 3x3x3 anti-symmetric tensor is separable.