A republic with an emperor?

Hi All,

Some of my American friends have gripes over me calling some of my science fiction governments republics when they are led by an emperor.

This is understandable, because the current definition of a republic is simply a state that has no hereditary chain of succession. Thus the USA is a republic , so the vice president takes over if a president dies and not the first daughter or son.

However way back in history , we have the Romans. Historians like to divide Roman history into its republic and empire days. If you  have done enough pub quizzes  you might be able to say the first Roman Emperor was Augustus.

Yet Octavian , as was his actual name, politically claimed that he was merely restoring the Roman Republic and not abolishing it. Therefore throughout the rest of those imperial days, the leadership of Rome was sometimes inherited, in times of political stability , whereas others the army or Senate decided who was going to rule. This was true of the Eastern Roman " Byzantine"  half as well, with official, but not in fact non hereditary principle in place (sometimes the emperors would appoint their sons as co rulers to help a hereditary succession ) .

When one of the Popes made Charlemagne "Emperor of the Romans" , to piss off and power grab from the actual Romans who had a female Emperor (Byzantine was considered to be the Roman Empire and was called such, Byzantine was not a contemporary usage for the state) creating the Holy Roman Empire , the Imperial crown was not passed on a hereditary basis, but chosen by electors of this Confederate Empire.