A b s t r a c t. Let $H$ be the Hilbert-style intuitionistic propositional calculus without exchange and contraction rules (as given by Ono and Komori). An axiomatization of H with the separation property is provided. Of the superimplicational fragments of H, it is proved that just two fail to be finitely axiomatized, and that all are algebraizable. The paper is a study of these fragments, their equivalent algebraic (quasivariety) semantics and their axiomatic extensions.