History of the Roman Empire
Theodosius was the last Emperor who ruled over the whole Empire. After his death in 395, he gave the two halves of the Empire to his two sons Arcadius and Honorius; Arcadius became ruler in the East, with his capital in Constantinople, and Honorius became ruler in the West, with his capital in Milan and later Ravenna. The Roman state would continue to have two different emperors with different seats of power throughout the 5th century, though the Eastern Romans considered themselves to be the only ones who were fully Roman. Latin was used in official writings as much as, if not more than, Greek and the two halves were nominally, culturally and historically, if not politically, the same state.
Western Roman Empire
Charlemagne, and the subsequent Holy Roman Emperors, were not, and did not claim to be, rulers of a restored Western Roman Empire. Pope Leo III and contemporary historians were fully aware of that the notion of a separate Western court had been abolished over three centuries prior and considered the Roman Empire to be "one and indivisible". The ruler of the Roman Empire at the time of Charlemagne's coronation was Irene, the mother of emperor Constantine VI who she had deposed. Leo III considered Irene to be a usurper and illegitimate to rule due to her gender and as such considered the imperial throne to be vacant. Thus, Charlemagne was not crowned as the ruler of the Western Roman Empire and successor to Romulus Augustulus, but rather as the successor of Constantine VI and as sole Roman Emperor. Irene was deposed and replaced by Emperor Nikephoros soon after, and the Eastern Empire refused to recognize the Imperial title of Charlemagne. Following several wars in the 810s Emperor Michael I Rangabe eventually recognized Charlemagne as an "Emperor", but as the slightly humiliating "Emperor of the Franks" rather than "Roman Emperor", a title he reserved for himself. For centuries to come, the "revived" Western court and the Eastern court, in direct succession to the Roman Emperors of old, would make competing claims to be rulers of the whole Roman Empire. With the Eastern Empire terming the Holy Roman Empire as an "Empire of the Franks", the term "Empire of the Greeks" was popularized in the Frankish court as a way to refer to the Empire centered in Constantinople.