For a vector lattice E and n∈N\documentclass[12pt]{minimal}
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\begin{document}$$n \in \mathbb {N}$$\end{document}, let ⊗¯n,sE\documentclass[12pt]{minimal}
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\begin{document}$${\bar{\otimes }}_{n,s}E$$\end{document} denote the n-fold Fremlin vector lattice symmetric tensor product of E. For m,n∈N\documentclass[12pt]{minimal}
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\begin{document}$$m, n \in \mathbb {N}$$\end{document} with m>n\documentclass[12pt]{minimal}
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\begin{document}$$m > n$$\end{document}, we prove that (i) if ⊗¯m,sE\documentclass[12pt]{minimal}
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\begin{document}$${\bar{\otimes }}_{m,s}E$$\end{document} is uniformly complete then ⊗¯n,sE\documentclass[12pt]{minimal}
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\begin{document}$${\bar{\otimes }}_{n,s}E$$\end{document} is positively isomorphic to a complemented subspace of ⊗¯m,sE\documentclass[12pt]{minimal}
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\begin{document}$${\bar{\otimes }}_{m,s}E$$\end{document}, and (ii) if there exists [inline-graphic not available: see fulltext] such that ker(ϕ)\documentclass[12pt]{minimal}
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\begin{document}$$\ker (\phi )$$\end{document} is a projection band in E then ⊗¯n,sE\documentclass[12pt]{minimal}
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\begin{document}$${\bar{\otimes }}_{n,s}E$$\end{document} is lattice isomorphic to a projection band of ⊗¯m,sE\documentclass[12pt]{minimal}
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\begin{document}$${\bar{\otimes }}_{m,s}E$$\end{document}. We also obtain analogous results for the n-fold Fremlin projective symmetric tensor product ⊗^n,s,|π|E\documentclass[12pt]{minimal}
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\begin{document}$${\hat{\otimes }}_{n,s,|\pi |}E$$\end{document} of E where E is a Banach lattice.
