A republican democracy

Democracy

Democracy is generally defined as a form of government in which all adult citizens have an equal say in the decisions that affect their lives. Ideally, this includes equal participation in the proposal, development and passage of legislation into law...

is a republic

Republic

A republic is a form of government in which the people, or some significant portion of them, have supreme control over the government and where offices of state are elected or chosen by elected people. In modern times, a common simplified definition of a republic is a government where the head of...

 which has democratic

Democracy

Democracy is generally defined as a form of government in which all adult citizens have an equal say in the decisions that affect their lives. Ideally, this includes equal participation in the proposal, development and passage of legislation into law...

 forms of government. One of the key principles is free and open debate prior to casting a vote. A republic in the modern understanding is a nation

Nation

A nation may refer to a community of people who share a common language, culture, ethnicity, descent, and/or history. In this definition, a nation has no physical borders. However, it can also refer to people who share a common territory and government irrespective of their ethnic make-up...

 or state

State (polity)

A state is an organized political community, living under a government. States may be sovereign and may enjoy a monopoly on the legal initiation of force and are not dependent on, or subject to any other power or state. Many states are federated states which participate in a federal union...

 where the people are sovereign

Sovereignty

Sovereignty is the quality of having supreme, independent authority over a geographic area, such as a territory. It can be found in a power to rule and make law that rests on a political fact for which no purely legal explanation can be provided...

. It is not a monarchy

Monarchy

A monarchy is a form of government in which the office of head of state is usually held until death or abdication and is often hereditary and includes a royal house. In some cases, the monarch is elected...

, where the king or queen is the head of state

Head of State

A head of state is the individual that serves as the chief public representative of a monarchy, republic, federation, commonwealth or other kind of state. His or her role generally includes legitimizing the state and exercising the political powers, functions, and duties granted to the head of...

. By this definition there are abundant examples of states that are republics but not democracies, and of states that are democracies but not republics.