Webster's Twentieth Century Dictionary:
republic, n, ...
1. (a) a state or nation in which the supreme power rests in all the citizens entitled to vote (the electorate) and is exercised by representatives elected, directly or indirectly, by them and responsible to them; (b) the form of government of such a state or nation.
The founders of the Constitution laid the cornerstone of our national republic. -- Joseph Story.
2. ...< snip >...
3. a state or nation with a president as its titular head; distinguished from monarchy.
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U.S. Constitution, ARTICLE IV, SECTION 4:
"The United States shall guarantte to every State in this Union a Republican Form of Government, and shall protect each of them against Invasion; and on Application of the Legislature, or of the Executive (when the Legislature cannot be convened) against domestic Violence."