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For a positive integer n and a uniformly complete vector lattice E, let 
denote the n-fold Fremlin vector lattice symmetric tensor product of E. We prove that if there exists a lattice homomorphism φ ∈ E∼ then E is lattice isomorphic to a complemented sublattice of 
. Moreover, if ker(φ) is a projection band in E then the image of E is also a projection band in 
.
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