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We prove that all Arens extensions of finite rank Riesz multimorphisms taking values in Archimedean Riesz spaces coincide and are Riesz multimorphisms. Partial results for arbitrary Riesz multimorphisms are obtained. We also prove that, for a class of Banach lattices F , which includes F = c 0 , ℓ p , c 0 (ℓ p ), ℓ p (c 0 ), ℓ p (ℓ s ), 1 < p, s < ∞, among many others, all Aron-Berner extensions of F -valued Riesz multimorphisms between Banach lattices are Riesz multimorphisms.

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