how would you define this type of empire?
Romano-Hellenistic
Byzantine Empire
While the Byzantine Empire had a multi-ethnic character during most of its history and preserved Romano-Hellenistic traditions, it became identified by its western and northern contemporaries with its increasingly predominant Greek element. 
which region was part of the eastern roman empire The extent fluctuated, but at its peak, it stretched from Byzantium up to Syria and Italy and North Africa,
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Byzantine Empire [SEP] Nomenclature
While the Byzantine Empire had a multi-ethnic character during most of its history and preserved Romano-Hellenistic traditions, it became identified by its western and northern contemporaries with its increasingly predominant Greek element. Western medieval sources also referred to the empire as the "Empire of the Greeks" (Latin: "Imperium Graecorum") and to its emperor as "Imperator Graecorum" (Emperor of the Greeks); these terms were used to distinguish it from the Holy Roman Empire that claimed the prestige of the classical Roman Empire in the West.