The correct answer is a caliphate.
A caliphate is a Muslim state under the rule of a caliph, who is generally a governor who rules with absolute powers. Caliphs base the legitimacy of their position in the belief that they are sucessors of Prophet Muhammad, and therefore the new leaders of the Muslim community.
During the medieval era there were three principal caliphates: the Rashidun Caliphate (632–661), the Umayyad Caliphate (661–750) and the Abbasid Caliphate (750–1258).