Invertible substitutions in logics with algebraic semantics equivalent to Product algebras

Product logic is considered one of the major truth-functional fuzzy propositional logics. Its semantics is given by the variety of Product algebras ${\mathbb{P}}$. In the hierarchy of fuzzy logics based on left-continuous t-norms there are a few logics whose algebraic semantics are varieties categorically equivalent with ${\mathbb{P}}$. For these logics we shall describe finitely generated free algebras and their group of automorphisms, that is, invertible substitutions.