Complete latticeability in vector-valued sequence spaces

Using a technique due to Jiménez–Rodríguez, first we prove the complete latticeability of the set of disjoint non-norm null weakly null sequences and of the set of disjoint non-norm null regular-polynomially null sequences in Banach lattices. Then we apply the mother vector technique to prove the complete latticeability of $$\lambda _\pi (E) \setminus \lambda _s(E)$$, which implies the complete latticeability of $$(\ell _p\widehat{\otimes }_{|\pi |}E)\setminus (\ell _p\widehat{\otimes }_{\pi }E)$$, where E is a Banach lattice and $$1 < p < \infty$$.