Continuous Additive Algebras and Injective Simulations of Synchronization Trees

The (in)equational properties of the least fixed point operation on(omega-)continuous functions on (omega-)complete partially ordered sets arecaptured by the axioms of (ordered) iteration algebras, or iterationtheories. We show that the inequational laws of the sum operation inconjunction with the least fixed point operation in continuous additivealgebras have a finite axiomatization over the inequations of orderediteration algebras. As a byproduct of this relative axiomatizability result, we obtain complete infinite inequational and finite implicationalaxiomatizations. Along the way of proving these results, we give a concrete description of the free algebras in the corresponding variety ofordered iteration algebras. This description uses injective simulations of regular synchronization trees. Thus, our axioms are also sound andcomplete for the injective simulation (resource bounded simulation) of(regular) processes.Keywords: equational logic, fixed points, synchronization trees, simulation