About quasivarieties of p-algebras and Wajsberg algebras
A counterexample is given to show that not all quasivarieties of p-algebras lie between two consecutive varieties. It is shown that the quasivariety of p-algebras generated by the finite subdirectly irreducible p-algebras is the entire variety of p-algebras. Also, it is shown that this variety is not structurally complete and the class of its finitely subdirectly irreducible members coincides with the class of its subdirectly irreducible ones. This later result is used to show that there are no strict relatively congruence distributive quasivarieties of p-algebras. Relatively congruence distributive quasivarieties of Wajsberg algebras are characterized. The relative congruence extension property in the classes of p-algebras and Wajsberg algebras is studied. It is proved that in the first class only quasivarieties which are varieties possess this property. In the second one, it is shown that a quasivariety which is relatively congruence distributive or generates a proper subvariety has relative congruence extension property if and only if it is a variety.