The rules for appointing the president and the head of government allow in some republics to appoint a president and a prime minister who have opposite political beliefs: in France, when the members of the ruling cabinet and the president come from opposing political factions, this situation is called cohabitation. James Madison overthrew the small-scale republicanism of the Enlightenment in Federalist No. 10 (1787). He argued that in any free society, people would naturally come together in groups to pursue common interests. While all groups pursue their desires assuming they are rooted in the “common good,” many may actually pose a threat to the freedom of others. Madison called these malign groups “factions”: Calvinism played an important role in the Republican revolts in England and the Netherlands. Like the city-states of Italy and the Hanseatic League, both were important trading centers, with a large merchant class profiting from trade with the New World. Much of the population in both regions also embraced Calvinism. During the Dutch uprising (from 1566), the Dutch Republic emerged from the rejection of Spanish rule from the Habsburgs. However, the country did not immediately adopt the republican form of government: in the formal declaration of independence (Act of Abjuration, 1581), the throne was vacant only by King Philip, and the Dutch magistrates asked the Duke of Anjou, Queen Elizabeth of England and Prince William of Orange, one by one, to replace Philip. It was not until 1588 that the states (the state, the representative assembly at the time) decided to transfer the sovereignty of the country to each other. Terms such as “liberal republic” are also used to describe all modern liberal democracies.
 The people elect representatives to enact laws within the constraints of a constitution. In the years following World War II, most of the remaining European colonies gained independence and most became republics. The two largest colonial powers were France and the United Kingdom. Republican France encouraged the establishment of republics in its former colonies. The United Kingdom sought to follow the model it had for its former settlements of creating empires independent of the Commonwealth that were still united under the same monarch. While most settlements and small Caribbean states retained this system, it was rejected by newly independent countries in Africa and Asia, which revised their constitutions and became republics instead. Republicanism spread considerably after World War I, when several of Europe`s largest empires collapsed: the Russian Empire (1917), the German Empire (1918), the Austro-Hungarian Empire (1918), and the Ottoman Empire (1922) were all replaced by republics. New states gained independence during this turmoil, and many of them, such as Ireland, Poland, Finland and Czechoslovakia, chose republican forms of government. After Greece`s defeat in the Greco-Turkish War (1919–22), the monarchy was briefly replaced by the Second Hellenic Republic (1924–35). In 1931, the proclamation of the Second Spanish Republic (1931-39) led to the Spanish Civil War, which was to be the prelude to World War II. In liberal democracies, presidents are elected, either directly by the people or indirectly by a parliament or council. Typically, in presidential and semi-presidential systems, the president is elected directly by the people or indirectly elected, as is the case in the United States.
In this country, the president is formally elected by an electoral college elected by the states, all of which do so by direct election of voters. The indirect election of the President by the Electoral College corresponds to the concept of the republic as a unit with an indirect election system. According to some, direct elections give legitimacy to the president and give the office much of its political power.  However, this concept of legitimacy differs from that expressed in the U.S. Constitution, which establishes the legitimacy of the President of the United States following the signing of the Constitution by nine states.  The idea that direct elections are necessary for legitimacy also contradicts the spirit of the Great Compromise, the actual result of which was manifested in the clause giving voters in small states greater representation in presidential elections than in large states; For example, in 2016, Wyoming citizens had 3.6 times as many voters as California citizens.  The United States, like most modern nations, is neither a pure republic nor a pure democracy. Instead, it is a hybrid democratic republic. Republicanism would prove its political viability in the founding of America. The American founders attempted to introduce a form of democratic republicanism rather than pure democracy through the Constitution of 1787. What you should take away from the confusion (or debate) between democracy and republic is that in both forms of government, power ultimately belongs to the people who can vote.
If you have the right to vote, vote. This is what true democracies and republics are. It is sometimes argued that the former Soviet Union was also a supranational republic, based on the assertion that the Member States were different nation-states. However, in Book III of his Politics, Aristotle was apparently the first classical writer to state that the term politeia can be used to refer more specifically to a type of politeia: “When citizens generally rule for the common good, it is designated by the common name of all governments (to koinon onoma pasōn tōn politeiōn), Government (politeia)”. Even in classical Latin, the term “republic” can be used generally to refer to any regime, or in a particular way to designate governments that work for the common good.  The modern type of “republic” itself is different from any type of state found in the classical world.   Nevertheless, there are a number of Classical-era states that are still called republics today. These include ancient Athens and the Roman Republic. Although the structure and administration of these states differ from those of any modern republic, there is debate about the extent to which classical, medieval, and modern republics form a historical continuum. The genus is grouped in the genus “Homo”. Pocock argued that a distinct republican tradition extends from the classical world to the present day.
  Other researchers disagree.  Paul Rahe, for example, argues that classical republics had a form of government with little connection to those of a modern country.  The term republic does not appear in the Declaration of Independence, but it appears in Article IV of the Constitution, which “guarantees to each state of this union a republican form of government.” What exactly the framers of the Constitution thought this meant is uncertain. The Supreme Court, in Luther v. Borden (1849) explains that the definition of the republic is a “political question” in which it will not intervene. In two subsequent cases, a basic definition was established. In United States v. Cruikshank (1875), the court held that “equality of citizens” is inherent in the idea of a republic. Significantly, a key feature of the American constitutional republic also contradicts what many Enlightenment philosophers theorized as a necessary condition for a Republican government: a small territory. Montesquieu, for example, said: “It is natural that a republic should have only a small territory; Otherwise, it cannot survive for long. In a vast republic, the public good is sacrificed to a thousand private views; It is subject to exceptions and depends on accidents.
In one small case, the public interest is more obvious, better understood and more accessible to all citizens; Abuses are less extensive and, of course, less protected” (Baron de Montesquieu, Esprit de la Loi, 1748). You see, many democracies today are also republics and are even called democratic republics. The United States and France are therefore considered both democracies and republics – both terms indicate the fact that governmental power belongs to the people and that the exercise of that power is through some kind of electoral representation. What, then, is a republic? A republican government is one in which the people – directly or indirectly – are the ultimate source of authority and elect representatives to enact laws that serve their interests and promote the common good. But a constitutional republic also limits the power of the majority through a framework that fosters competent government and protects fundamental rights. You probably listen to countries like the United States or France, which are called democracies. At the same time, you probably also mean the two countries called republics. Is that possible? Are democracies and republics the same or different? The Bill of Rights is perhaps the clearest example of the dichotomy between a democracy and a republic. It is at the same time the most famous feature of our “democracy” and the most anti-democratic feature of our constitutional republic.