Complementation in the Fremlin vector lattice symmetric tensor products-II

For a vector lattice E and n is an element of N, let (circle times) over bar (n,s) E denote the n-fold Fremlin vector lattice symmetric tensor product of E. For m, n is an element of N with m > n, we prove that (i) if (circle times) over bar (m,s) E is uniformly complete then (circle times) over bar (n,s) E is positively isomorphic to a complemented subspace of (circle times) over bar (m,s) E, and (ii) if there exists phi is an element of E-+(similar to) such that ker(phi) is a projection band in E then (circle times) over bar (n,s) E is lattice isomorphic to a projection band of (circle times) over bar (m,s) E. We also obtain analogous results for the n-fold Fremlin projective symmetric tensor product (circle times) over cap (n,s,vertical bar pi vertical bar) E of E where E is a Banach lattice.