Morita Rigidity for Kleene Algebras

We introduce Morita equivalence to the study of Kleene algebras and modules. Classical characterizations of Morita-equivalent semirings such as having equivalent categories of modules and one semiring being a full matrix algebra over the other carry over. We also observe that Morita equivalence can be applied to extending and restricting scalars in Lindenbaum Tarski algebras of propositional dynamic logics. But the signature result which we obtain is a form of rigidity for Kleene algebras, which states that if the semiring reducts of two Kleene algebras are Morita-equivalent, then the Morita equivalence is in fact witnessed by Kleene bimodules.