Public services such as government, legislation police or the military will be required on Mars as on Earth. Public service facilities may be a simple office for an outpost, or a full fledged building for a large self sufficient colony. The separation of powers is a common model for the organisation of a community or state. A system of checks and balances is set in place to prevent the risk of autocracy. Models may include more or less branches than the standard three branch model.
Au auditing branch may be explicitly added to the structure, for example.
A martian settlement, or colony, may be subsidiary to Earth authorities or it may exercise full authority.
Public services eventually require facilities to carry out their responsibilities. Police stations, fire stations, court houses, legislature buildings are all eventually required for Mars. The military may require extensive facilities or it may be practically non-existent, depending on conditions and colony development.
In many countries public services extend to infrastructure, health services and education. A notable exception is the United States. The nature of public services on Mars is likely to be an early debate in the development of the settlement, and is also likely to change over time.