Morita Rigidity for Kleene Algebras

TL;DR

This paper explores Morita equivalence in Kleene algebras, revealing a rigidity property that links Morita equivalence of their semiring reducts to Kleene bimodules, advancing algebraic understanding of these structures.

Contribution

It introduces Morita equivalence to Kleene algebras and establishes a rigidity result connecting Morita equivalence of reducts to Kleene bimodules.

Findings

Morita equivalence extends to Kleene algebras and modules.

Morita equivalence of semiring reducts implies bimodule witness.

Rigidity result links Morita equivalence to Kleene bimodules.

Abstract

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.