Structurally complete finitary extensions of positive Łukasiewicz logic

In this paper we study ${\mathcal{M}\mathcal{V}}^{+}$, i.e. the positive fragment of Łukasiewicz Multi-Valued Logic $\mathcal{M}\mathcal{V}$. In particular we describe all the finitary extensions of ${\mathcal{M}\mathcal{V}}^{+}$ that are structurally complete and all the axiomatic extensions of ${\mathcal{M}\mathcal{V}}^{+}$ that are hereditarily structurally complete. Examples of hereditarily structurally complete finitary extensions and non hereditarily structurally complete finitary extensions are provided.