Expanding the Leibniz Hierarchy

In this work we present two new classes of logics determined by specific properties of the Leibniz operator that expand the Leibniz hierarchy. By doing so, we present a sufficient condition for the commutativity of the Leibniz operator with inverse substitutions/homomorphisms, which gives us a negative answer to the possibility of establishing a precise connection between Leibniz classes and the behavior of the Leibniz operator.