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.

翻译：我们将Morita等价引入Kleene代数与模的研究。关于Morita等价半环的经典刻画——例如具有等价的模范畴，以及一个半环是另一个半环上的全矩阵代数——在此情形下依然成立。我们还观察到，Morita等价可应用于命题动态逻辑的Lindenbaum Tarski代数中的标量扩张与限制。但我们获得的核心结果是一种Kleene代数的刚性形式，该结果表明：若两个Kleene代数的半环约化是Morita等价的，则该Morita等价实际上可由Kleene双模实现。