Bases in the space of regular multilinear operators on Banach lattices

For Banach lattices E1,…,Em and F with 1-unconditional bases, we show that the monomial sequence forms a 1-unconditional basis of Lr(E1,…,Em;F), the Banach lattice of all regular m-linear operators from E1×⋯×Em to F, if and only if each basis of E1,…,Em is shrinking and every positive m-linear operator from E1×⋯×Em to F is weakly sequentially continuous. As a consequence, we obtain necessary and sufficient conditions for which the m-fold Fremlin projective tensor product E1⊗ˆ|π|⋯⊗ˆ|π|Em (resp. the m-fold positive injective tensor product E1⊗ˇ|ϵ|⋯⊗ˇ|ϵ|Em) has a shrinking basis or a boundedly complete basis.