Etymology
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polemarch (n.)

"commander of the army," a title of certain officers in Greek history, 1570s, from Greek polemarkhos "one who begins or leads a war," from polemos "war" (a word of unknown origin) + arkhos "leader, chief, ruler" (see archon).

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-arch 

word-forming element meaning "a ruler," from Greek arkhos "leader, chief, ruler," from arkhē "beginning, origin, first place," verbal noun of arkhein "to be the first," hence "to begin" and "to rule" (see archon).

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-archy 

word-forming element meaning "rule," from Latin -archia, from Greek -arkhia "rule," from arkhos "leader, chief, ruler," from arkhē "beginning, origin, first place," verbal noun of arkhein "to be the first," hence "to begin" and "to rule" (see archon).

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arch- 
also archi-, word-forming element meaning "chief, principal; extreme, ultra; early, primitive," from Latinized form of Greek arkh-, arkhi- "first, chief, primeval," combining form of arkhos "a chief, leader, commander," arkhein "be first, begin" (see archon).
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exarch (n.)

historically, "a ruler of a province in the Byzantine Empire;" in the early Church, "a prelate presiding over a diocese;" in the Greek Church, a legate of a patriarch; from Late Latin exarchus, from Greek exarkhos "a leader," from ex (see ex-) + arkhos "leader, chief, ruler" (see archon). Related: Exarchate.

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anarchy (n.)

1530s, "absence of government," from French anarchie or directly from Medieval Latin anarchia, from Greek anarkhia "lack of a leader, the state of people without a government" (in Athens, used of the Year of Thirty Tyrants, 404 B.C., when there was no archon), abstract noun from anarkhos "rulerless," from an- "without" (see an- (1)) + arkhos "leader" (see archon).

From 1660s as "confusion or absence of authority in general;" by 1849 in reference to the social theory advocating "order without power," with associations and co-operatives taking the place of direct government, as formulated in the 1830s by French political philosopher Pierre-Joseph Proudhon (1809-1865).

Either the State for ever, crushing individual and local life, taking over in all fields of human activity, bringing with it its wars and its domestic struggles for power, its palace revolutions which only replace one tyrant by another, and inevitably at the end of this development there is ... death! Or the destruction of States, and new life starting again in thousands of centers on the principle of the lively initiative of the individual and groups and that of free agreement. The choice lies with you! [Prince Peter Kropotkin (1842-1921), "The State: Its Historic Role," 1896]
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