A while back I learned that even after the rule of Augustus and the foundation of the Empire, the citizens of Rome continued to call it and consider it a continuation of the Republic, and that the separation between the two is a modern invention.
However, lately I've also been reading on the Byzantine/Eastern Roman Empire and now I'm wondering if they still considered themselves to be the 'Roman Republic' rather than a proper empire? Or were any notions of preserving the Republic gone by the time of the Dominate?
I'll refer you to an answer I gave to a similar question. The basic answer is not to project our modern conception of a republic (representative democracy) back into Roman/Byzantine times.