Abstract
In this paper we deal with quasivarieties of residuated structures which form the equivalent algebraic semantics of a positive fragment of some substructural logic. Our focus is mainly on varieties and quasivarieties of Wajsberg hoops, which are the equivalent algebraic semantics of the positive fragment of Łukasiewicz many-valued logic. In particular we study the lattice of subquasivarieties of Wajsberg hoops and we describe completely all the subvarieties of Wajsberg hoops that are primitive. Though the treatment is mostly algebraic in nature, there are obvious connections with the underlying logics.
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