empire (n.)youdaoicibaDictYouDict
early 14c., from Old French empire "rule, authority, kingdom, imperial rule" (11c.), from Latin imperium "a rule, a command; authority, control, power; supreme power, sole dominion; military authority; a dominion, realm," from imperare "to command," from assimilated form of in- "in" (see in- (2)) + parare "to order, prepare" (see pare).
[P]roperly an empire is an aggregate of conquered, colonized, or confederated states, each with its own government subordinate or tributary to that of the empire as a whole. [Century Dictionary]
Not etymologically restricted to "territory ruled by an emperor," but used that way. The Empire, meaning "the British Empire," first recorded 1772 (it officially devolved into "The Commonwealth" in 1931); before that it meant the Holy Roman Empire (1670s). Empire as the name of a style (especially in reference to a style of dresses with high waistlines) is by 1869, in reference to the affected classicism prevailing in France during the reign of Napoleon I (1804-15). Second Empire is in reference to the rule of Napoleon III of France (1852-70). New York has been called the Empire State since 1834.
fasces (n.)youdaoicibaDictYouDict
1590s, from Latin fasces "bundle of rods containing an axe with the blade projecting" (plural of fascis "bundle" of wood, etc.), from Proto-Italic *faski- "bundle," perhaps from PIE *bhasko- "band, bundle" (cognates: Middle Irish basc "neckband," Welsh baich "load, burden," perhaps also Old English bæst "inner bark of the linden tree"). Carried before a lictor, a superior Roman magistrate, as a symbol of power over life and limb: the sticks symbolized punishment by whipping, the axe-head execution by beheading. Hence in Latin it also meant, figuratively, "high office, supreme power."
republic (n.)youdaoicibaDictYouDict
c. 1600, "state in which supreme power rests in the people via elected representatives," from Middle French république (15c.), from Latin respublica (ablative republica) "the common weal, a commonwealth, state, republic," literally res publica "public interest, the state," from res "affair, matter, thing" + publica, fem. of publicus "public" (see public (adj.)). Republic of letters attested from 1702.