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