Splittings and Finite Basis Theorems

We study splittings of a lattice, first from a general lattice-theoretic point of view, then applying the obtained results to complete lattices of classes of algebras (mostly quasivarieties and varieties). In particular, the notion of separability of lattice, which entails many known properties of splittings in latices of varieties and quasivarieties alike, is of particular importance. We point out many differences between properties of splittings in lattices of subvarieties and subquasivarieties. We also introduce the notions of weak primitivity and weak tameness and we prove the finite basis theorem for weakly primitive weakly tame quasivarieties of finite type.