Multiplicative intuitionistic linear logic (MILL) has only multiplicative conjunction $\otimes$ and linear implication $\multimap$ as connectives. It has models in symmetric monoidal closed categories.
Compact closed categories are symmetric monoidal closed categories in which every object $A$ has a dual $A^*$ and $A \multimap B \cong A^* \otimes B$. Thought of as a resource, $A^*$ is a debt, owing someone an $A$. Is there a special name for MILL when these conditions hold?