The first French Republic, established after the revolution, was of course politically the inheritor of the Kingdom of France and so ultimately of Clovis' and Charlemagne's Gallic 'Roman Empire', but it also deliberately set itself up to evoke the ancient Roman Republic. The trappings of central government: consuls, prefects, the senate etc.; the symbols of the state such as, the goddess Liberty, aka 'Marianne', the fasces, eagles on military standards etc.; and culturally things such as the so-called Directoire and Empire styles of architecture and furniture, and classical subjects in painting and sculpture; were all deliberately made to recall the style, trappings, and of course imperial might, of ancient Rome.
PS from wiki: The historian Hugh Honour in 'Neo-classicism. Style and Civilisation' (1968) wrote of the Directory period,
"The stoic virtues of Republican Rome were upheld as standards not merely for the arts but also for political behaviour and private morality. Conventionels saw themselves as antique heroes. Children were named after Brutus, Solon and Lycurgus. The festivals of the Revolution were staged by Jacques-Louis David as antique rituals. Even the chairs in which the committee of Salut Publique sat were made on antique models devised by David.... In fact Neo-classicism became fashionable".
But how much did they actually believe themselves to be the true descendents and inheritors of the Roman Empire?