On the quasivariety of BCK-algebras and its subvarieties (Q1344842)

The quasivariety \({\mathcal B}{\mathcal C}{\mathcal K}\) of all BCK-algebras is the splitting quasivariety associated with a certain 3-element algebra \({\mathbf B}\) in a large variety \({\mathcal V}\), i.e. \({\mathcal B}{\mathcal C}{\mathcal K}\) consists of all algebras in \(\mathcal V\) that do not contain a subalgebra isomorphic to \({\mathbf B}\). Moreover, the varieties of BCK-algebras (i.e. varieties contained in \({\mathcal B}{\mathcal C}{\mathcal K})\) form a sublattice of the lattice of subquasivarieties of \({\mathcal B}{\mathcal C}{\mathcal K}\) and this sublattice contains a cofinal chain of order type \(\omega\), but no chain of quasicommutative BCK-varieties is cofinal among the varieties of BCK- algebras; neither arises a natural chain of varieties from Cornish's condition (J). Also some new examples of BCK-algebras are given.