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In this paper we construct Darboux transformations and Bäcklund transformations for the supersymmetric two-boson hierarchy. Three Darboux transformations and associated Bäcklund transformations are obtained. For one of them, we also work out the corresponding nonlinear superposition formula. We build, for both the supersymmetric two-boson equation and the N = 2 , a = 4 supersymmetric KdV equation, their one-soliton solutions and present their semi-discrete versions. [ABSTRACT FROM AUTHOR]