Lattices of boundedly axiomatizable $\forall$-subclasses of $\forall$-classes of universal algebras

The question about the structure of lattices of subclasses of various classes of algebras is one of the basic ones in universal algebra. The case under consideration most frequently concerns lattices of subvarieties (subquasivarieties) of varieties (quasivarieties) of universal algebras. A similar question is also meaningful for other classes of algebras, in particular, for universal classes of algebras. The union of two $\forall$-classes is itself a $\forall$-class, hence such lattices are distributive. As a rule, those lattices of subclasses are rather large and are not simply structured. In this connection, it is of interest to distinguish some sublattices of such lattices that would model certain properties of the lattices themselves. The present paper deals with a similar problem for $\forall$-classes and varieties of universal algebras.