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Let V be a vertex operator algebra. The fusion products in this paper are defined by logarithmic intertwining operators. Under this setting, we prove the pure exactness of any extension of the V -module V . Namely, if 0 → Q → ϵ P → ρ V → 0 is a short exact sequence of V -modules and W is a V -module, then 0 → Q ⊠ W → ϵ ⊠ 1 W P ⊠ W → ρ ⊠ 1 W V ⊠ W → 0 is also exact, where we view it as a sequence of g ( V ) -modules. [ABSTRACT FROM AUTHOR]

Abstract: A duality of two categories is introduced. It generalizes the Malcev description of torsion free finite rank abelian groups. An equivalence of two categories is also introduced. It generalizes the Kurosh description of p-primitive groups. The composition of the duality and the equivalence is the duality earlier introduced by W. Wickless and the author. It is shown that the last duality preserves exactness for short exact sequences of homomorphisms. [Copyright &y& Elsevier]