One-sorted Program Algebras

Kleene algebra with tests, KAT, provides a simple two-sorted algebraic framework for verifying properties of propositional while programs. Kleene algebra with domain, KAD, is a one-sorted alternative to KAT. The equational theory of KAT embeds into KAD, but KAD lacks some natural properties of KAT. For instance, not each Kleene algebra expands to a KAD, and the subalgebra of tests in each KAD is forced to be the maximal Boolean subalgebra of the negative cone. In this paper we propose a generalization of KAD that avoids these features while still embedding the equational theory of KAT. We show that several natural properties of the domain operator of KAD can be added to the generalized framework without affecting the results. We consider a variant of the framework where test complementation is defined using a residual of the Kleene algebra multiplication.