Kleene Algebrae, Kleene Modules, and Morita Equivalence
Luke Serafin
TL;DR
The paper extends Morita theory to Kleene algebrae by developing Kleene modules, duals, tensor products, and a Morita category. It introduces the tensor product of Kleene modules via a free construction and a congruence to realize adjunction with $\mathrm{Hom}$, and proves key isomorphisms that underpin Morita equivalence in this setting. It shows how homomorphism modules $E_h$ behave under composition and demonstrates that Morita equivalence corresponds to equivalence of module categories, with matrix algebras $M_n(K)$ Morita-equivalent to $K$, via the bimodules $K^n$ and $K^{n\circ}$. The results link automata-theoretic behavior expressed over Kleene algebras to algebraic Morita theory and suggest a robust framework for further structural study.
Abstract
Modules and the notion of Morita equivalence are foundational to the classical study of rings. These concepts extend naturally to semirings and then specialize to Kleene algebrae, and my goal is to investigate Kleene modules and Morita equivalence of Kleene algebrae in the hope that some of the power seen in the context of rings may be found in this new context as well.